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Introduction
In this article, we will explore the concept of probability and its application in a multiple-choice quiz scenario. A quiz consists of 20 multiple-choice questions, each with 5 possible answers. We will calculate the probability of passing the quiz if the minimum passing grade is 60 % 60\% 60%
Understanding the Problem
To begin with, let's understand the problem at hand. We have a quiz with 20 multiple-choice questions, each with 5 possible answers. This means that for each question, there are 5 possible outcomes, and the individual making random guesses has an equal chance of selecting any of the 5 options.
Calculating the Probability of Passing
To calculate the probability of passing the quiz, we need to determine the number of correct answers required to achieve a minimum passing grade of 60 % 60\% 60% 60 % 60\% 60%
Probability of Getting a Single Question Correct
Now, let's calculate the probability of getting a single question correct. Since there are 5 possible answers, the probability of selecting the correct answer is 1 out of 5, which is 1 5 \frac{1}{5} 5 1 β
Probability of Getting a Single Question Incorrect
Similarly, the probability of getting a single question incorrect is 4 out of 5, which is 4 5 \frac{4}{5} 5 4 β
Probability of Getting Exactly 12 Questions Correct
To calculate the probability of getting exactly 12 questions correct, we need to use the binomial probability formula. The binomial probability formula is given by:
P ( X = k ) = ( n k ) p k ( 1 β p ) n β k P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
P ( X = k ) = ( k n β ) p k ( 1 β p ) n β k
where n n n k k k p p p ( n k ) \binom{n}{k} ( k n β )
In this case, n = 20 n = 20 n = 20 k = 12 k = 12 k = 12 p = 0.2 p = 0.2 p = 0.2
P ( X = 12 ) = ( 20 12 ) ( 0.2 ) 12 ( 0.8 ) 8 P(X = 12) = \binom{20}{12} (0.2)^{12} (0.8)^{8}
P ( X = 12 ) = ( 12 20 β ) ( 0.2 ) 12 ( 0.8 ) 8
Calculating the Binomial Coefficient
The binomial coefficient ( 20 12 ) \binom{20}{12} ( 12 20 β )
( n k ) = n ! k ! ( n β k ) ! \binom{n}{k} = \frac{n!}{k!(n-k)!}
( k n β ) = k ! ( n β k )! n ! β
where n ! n! n ! n n n
( 20 12 ) = 20 ! 12 ! ( 20 β 12 ) ! = 20 ! 12 ! 8 ! \binom{20}{12} = \frac{20!}{12!(20-12)!} = \frac{20!}{12!8!}
( 12 20 β ) = 12 ! ( 20 β 12 )! 20 ! β = 12 ! 8 ! 20 ! β
Evaluating the Factorials
To evaluate the factorials, we can use the following formula:
n ! = n Γ ( n β 1 ) Γ ( n β 2 ) Γ β¦ Γ 2 Γ 1 n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1
n ! = n Γ ( n β 1 ) Γ ( n β 2 ) Γ β¦ Γ 2 Γ 1
Plugging in the values, we get:
20 ! = 20 Γ 19 Γ 18 Γ β¦ Γ 2 Γ 1 20! = 20 \times 19 \times 18 \times \ldots \times 2 \times 1
20 ! = 20 Γ 19 Γ 18 Γ β¦ Γ 2 Γ 1
12 ! = 12 Γ 11 Γ 10 Γ β¦ Γ 2 Γ 1 12! = 12 \times 11 \times 10 \times \ldots \times 2 \times 1
12 ! = 12 Γ 11 Γ 10 Γ β¦ Γ 2 Γ 1
8 ! = 8 Γ 7 Γ 6 Γ β¦ Γ 2 Γ 1 8! = 8 \times 7 \times 6 \times \ldots \times 2 \times 1
8 ! = 8 Γ 7 Γ 6 Γ β¦ Γ 2 Γ 1
Calculating the Binomial Coefficient
Now that we have evaluated the factorials, we can calculate the binomial coefficient:
( 20 12 ) = 20 ! 12 ! 8 ! = 2432902008176640000 479001600 = 125970 \binom{20}{12} = \frac{20!}{12!8!} = \frac{2432902008176640000}{479001600} = 125970
( 12 20 β ) = 12 ! 8 ! 20 ! β = 479001600 2432902008176640000 β = 125970
Calculating the Probability of Getting Exactly 12 Questions Correct
Now that we have calculated the binomial coefficient, we can calculate the probability of getting exactly 12 questions correct:
P ( X = 12 ) = ( 20 12 ) ( 0.2 ) 12 ( 0.8 ) 8 = 125970 Γ ( 0.2 ) 12 Γ ( 0.8 ) 8 P(X = 12) = \binom{20}{12} (0.2)^{12} (0.8)^{8} = 125970 \times (0.2)^{12} \times (0.8)^{8}
P ( X = 12 ) = ( 12 20 β ) ( 0.2 ) 12 ( 0.8 ) 8 = 125970 Γ ( 0.2 ) 12 Γ ( 0.8 ) 8
Evaluating the Exponents
To evaluate the exponents, we can use the following formula:
a b = a Γ a Γ β¦ Γ a β b Β times a^b = \underbrace{a \times a \times \ldots \times a}_{b \text{ times}}
a b = b Β times a Γ a Γ β¦ Γ a β β
Plugging in the values, we get:
( 0.2 ) 12 = 0.2 Γ 0.2 Γ β¦ Γ 0.2 Β (12Β times) (0.2)^{12} = 0.2 \times 0.2 \times \ldots \times 0.2 \text{ (12 times)}
( 0.2 ) 12 = 0.2 Γ 0.2 Γ β¦ Γ 0.2 Β (12Β times)
( 0.8 ) 8 = 0.8 Γ 0.8 Γ β¦ Γ 0.8 Β (8Β times) (0.8)^{8} = 0.8 \times 0.8 \times \ldots \times 0.8 \text{ (8 times)}
( 0.8 ) 8 = 0.8 Γ 0.8 Γ β¦ Γ 0.8 Β (8Β times)
Calculating the Probability of Getting Exactly 12 Questions Correct
Now that we have evaluated the exponents, we can calculate the probability of getting exactly 12 questions correct:
P(X = 12) = 125970 \times (0.2)^{12} \times (0.8)^{8} = 125970 \times 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000<br/>
# A Quiz with Random Guesses: Calculating the Probability of Passing
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Q&A: Calculating the Probability of Passing a Quiz with Random Guesses
Q: What is the probability of passing a quiz with 20 multiple-choice questions, each with 5 possible answers, if the minimum passing grade is 60 % 60\% 60%
A: To calculate the probability of passing the quiz, we need to determine the number of correct answers required to achieve a minimum passing grade of 60 % 60\% 60% 60 % 60\% 60%
Q: What is the probability of getting a single question correct?
A: Since there are 5 possible answers, the probability of selecting the correct answer is 1 out of 5, which is 1 5 \frac{1}{5} 5 1 β
Q: What is the probability of getting a single question incorrect?
A: Similarly, the probability of getting a single question incorrect is 4 out of 5, which is 4 5 \frac{4}{5} 5 4 β
Q: How do we calculate the probability of getting exactly 12 questions correct?
A: To calculate the probability of getting exactly 12 questions correct, we need to use the binomial probability formula. The binomial probability formula is given by:
P ( X = k ) = ( n k ) p k ( 1 β p ) n β k < / s p a n > < / p > < p > w h e r e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > n < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e n u m b e r o f t r i a l s , < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > k < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > k < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n β r i g h t : 0.03148 e m ; " > k < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e n u m b e r o f s u c c e s s e s , < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > p < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > p < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.625 e m ; v e r t i c a l β a l i g n : β 0.1944 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > p < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e p r o b a b i l i t y o f s u c c e s s , a n d < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c l i n e t h i c k n e s s = " 0 p x " > < m i > n < / m i > < m i > k < / m i > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > ( n k ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.2 e m ; v e r t i c a l β a l i g n : β 0.35 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7454 e m ; " > < s p a n s t y l e = " t o p : β 2.355 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n β r i g h t : 0.03148 e m ; " > k < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.144 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e b i n o m i a l c o e f f i c i e n t . < / p > < h 3 > Q : W h a t i s t h e b i n o m i a l c o e f f i c i e n t < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c l i n e t h i c k n e s s = " 0 p x " > < m n > 20 < / m n > < m n > 12 < / m n > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > ( 20 12 ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.2451 e m ; v e r t i c a l β a l i g n : β 0.35 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8951 e m ; " > < s p a n s t y l e = " t o p : β 2.355 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.144 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 20 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > ? < / h 3 > < h 3 > A : T h e b i n o m i a l c o e f f i c i e n t < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c l i n e t h i c k n e s s = " 0 p x " > < m n > 20 < / m n > < m n > 12 < / m n > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > ( 20 12 ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.2451 e m ; v e r t i c a l β a l i g n : β 0.35 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8951 e m ; " > < s p a n s t y l e = " t o p : β 2.355 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.144 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 20 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > c a n b e c a l c u l a t e d u s i n g t h e f o r m u l a : < / h 3 > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c l i n e t h i c k n e s s = " 0 p x " > < m i > n < / m i > < m i > k < / m i > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < m o > = < / m o > < m f r a c > < m r o w > < m i > n < / m i > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < m r o w > < m i > k < / m i > < m o s t r e t c h y = " f a l s e " > ! < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > n < / m i > < m o > β < / m o > < m i > k < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > ( n k ) = n ! k ! ( n β k ) ! < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v e r t i c a l β a l i g n : β 0.95 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.1076 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n β r i g h t : 0.03148 e m ; " > k < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ) < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.3074 e m ; v e r t i c a l β a l i g n : β 0.936 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n β r i g h t : 0.03148 e m ; " > k < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > β < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n β r i g h t : 0.03148 e m ; " > k < / s p a n > < s p a n c l a s s = " m c l o s e " > ) ! < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c β l i n e " s t y l e = " b o r d e r β b o t t o m β w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.936 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > w h e r e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > n < / m i > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > n ! < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e f a c t o r i a l o f < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > n < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > . P l u g g i n g i n t h e v a l u e s , w e g e t : < / p > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c l i n e t h i c k n e s s = " 0 p x " > < m n > 20 < / m n > < m n > 12 < / m n > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < m o > = < / m o > < m f r a c > < m r o w > < m n > 20 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < m r o w > < m n > 12 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 20 < / m n > < m o > β < / m o > < m n > 12 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < / m f r a c > < m o > = < / m o > < m f r a c > < m r o w > < m n > 20 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < m r o w > < m n > 12 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < m n > 8 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > ( 20 12 ) = 20 ! 12 ! ( 20 β 12 ) ! = 20 ! 12 ! 8 ! < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v e r t i c a l β a l i g n : β 0.95 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 20 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ) < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.3074 e m ; v e r t i c a l β a l i g n : β 0.936 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 20 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > β < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) ! < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c β l i n e " s t y l e = " b o r d e r β b o t t o m β w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 20 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.936 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0574 e m ; v e r t i c a l β a l i g n : β 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < s p a n c l a s s = " m o r d " > 8 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c β l i n e " s t y l e = " b o r d e r β b o t t o m β w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 20 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 3 > Q : H o w d o w e e v a l u a t e t h e f a c t o r i a l s ? < / h 3 > < h 3 > A : T o e v a l u a t e t h e f a c t o r i a l s , w e c a n u s e t h e f o l l o w i n g f o r m u l a : < / h 3 > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m i > n < / m i > < m o s t r e t c h y = " f a l s e " > ! < / m o > < m o > = < / m o > < m i > n < / m i > < m o > Γ < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > n < / m i > < m o > β < / m o > < m n > 1 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > Γ < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > n < / m i > < m o > β < / m o > < m n > 2 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > Γ < / m o > < m o > β¦ < / m o > < m o > Γ < / m o > < m n > 2 < / m n > < m o > Γ < / m o > < m n > 1 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > n ! = n Γ ( n β 1 ) Γ ( n β 2 ) Γ β¦ Γ 2 Γ 1 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > β < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > β < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > β¦ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > P l u g g i n g i n t h e v a l u e s , w e g e t : < / p > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m n > 20 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < m o > = < / m o > < m n > 20 < / m n > < m o > Γ < / m o > < m n > 19 < / m n > < m o > Γ < / m o > < m n > 18 < / m n > < m o > Γ < / m o > < m o > β¦ < / m o > < m o > Γ < / m o > < m n > 2 < / m n > < m o > Γ < / m o > < m n > 1 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > 20 ! = 20 Γ 19 Γ 18 Γ β¦ Γ 2 Γ 1 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 20 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 20 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 19 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 18 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > β¦ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m n > 12 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < m o > = < / m o > < m n > 12 < / m n > < m o > Γ < / m o > < m n > 11 < / m n > < m o > Γ < / m o > < m n > 10 < / m n > < m o > Γ < / m o > < m o > β¦ < / m o > < m o > Γ < / m o > < m n > 2 < / m n > < m o > Γ < / m o > < m n > 1 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > 12 ! = 12 Γ 11 Γ 10 Γ β¦ Γ 2 Γ 1 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 11 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 10 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > β¦ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m n > 8 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < m o > = < / m o > < m n > 8 < / m n > < m o > Γ < / m o > < m n > 7 < / m n > < m o > Γ < / m o > < m n > 6 < / m n > < m o > Γ < / m o > < m o > β¦ < / m o > < m o > Γ < / m o > < m n > 2 < / m n > < m o > Γ < / m o > < m n > 1 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > 8 ! = 8 Γ 7 Γ 6 Γ β¦ Γ 2 Γ 1 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 8 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 8 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 7 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 6 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > β¦ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 3 > Q : W h a t i s t h e v a l u e o f t h e b i n o m i a l c o e f f i c i e n t < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c l i n e t h i c k n e s s = " 0 p x " > < m n > 20 < / m n > < m n > 12 < / m n > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > ( 20 12 ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.2451 e m ; v e r t i c a l β a l i g n : β 0.35 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8951 e m ; " > < s p a n s t y l e = " t o p : β 2.355 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.144 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 20 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > ? < / h 3 > < h 3 > A : N o w t h a t w e h a v e e v a l u a t e d t h e f a c t o r i a l s , w e c a n c a l c u l a t e t h e b i n o m i a l c o e f f i c i e n t : < / h 3 > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c l i n e t h i c k n e s s = " 0 p x " > < m n > 20 < / m n > < m n > 12 < / m n > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < m o > = < / m o > < m f r a c > < m r o w > < m n > 20 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < m r o w > < m n > 12 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < m n > 8 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < / m f r a c > < m o > = < / m o > < m f r a c > < m n > 2432902008176640000 < / m n > < m n > 479001600 < / m n > < / m f r a c > < m o > = < / m o > < m n > 125970 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > ( 20 12 ) = 20 ! 12 ! 8 ! = 2432902008176640000 479001600 = 125970 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v e r t i c a l β a l i g n : β 0.95 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 20 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ) < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0574 e m ; v e r t i c a l β a l i g n : β 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < s p a n c l a s s = " m o r d " > 8 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c β l i n e " s t y l e = " b o r d e r β b o t t o m β w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 20 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v e r t i c a l β a l i g n : β 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 479001600 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c β l i n e " s t y l e = " b o r d e r β b o t t o m β w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2432902008176640000 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 125970 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 3 > Q : H o w d o w e c a l c u l a t e t h e p r o b a b i l i t y o f g e t t i n g e x a c t l y 12 q u e s t i o n s c o r r e c t ? < / h 3 > < h 3 > A : N o w t h a t w e h a v e c a l c u l a t e d t h e b i n o m i a l c o e f f i c i e n t , w e c a n c a l c u l a t e t h e p r o b a b i l i t y o f g e t t i n g e x a c t l y 12 q u e s t i o n s c o r r e c t : < / h 3 > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m i > P < / m i > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > X < / m i > < m o > = < / m o > < m n > 12 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c l i n e t h i c k n e s s = " 0 p x " > < m n > 20 < / m n > < m n > 12 < / m n > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 0.2 < / m n > < m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m n > 12 < / m n > < / m s u p > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 0.8 < / m n > < m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m n > 8 < / m n > < / m s u p > < m o > = < / m o > < m n > 125970 < / m n > < m o > Γ < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 0.2 < / m n > < m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m n > 12 < / m n > < / m s u p > < m o > Γ < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 0.8 < / m n > < m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m n > 8 < / m n > < / m s u p > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > P ( X = 12 ) = ( 20 12 ) ( 0.2 ) 12 ( 0.8 ) 8 = 125970 Γ ( 0.2 ) 12 Γ ( 0.8 ) 8 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n β r i g h t : 0.13889 e m ; " > P < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n β r i g h t : 0.07847 e m ; " > X < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v e r t i c a l β a l i g n : β 0.95 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : β 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 20 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ) < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 0.2 < / s p a n > < s p a n c l a s s = " m c l o s e " > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 0.8 < / s p a n > < s p a n c l a s s = " m c l o s e " > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 8 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 125970 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 0.2 < / s p a n > < s p a n c l a s s = " m c l o s e " > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 0.8 < / s p a n > < s p a n c l a s s = " m c l o s e " > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 8 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 3 > Q : W h a t i s t h e v a l u e o f t h e p r o b a b i l i t y o f g e t t i n g e x a c t l y 12 q u e s t i o n s c o r r e c t ? < / h 3 > < h 3 > A : N o w t h a t w e h a v e e v a l u a t e d t h e e x p o n e n t s , w e c a n c a l c u l a t e t h e p r o b a b i l i t y o f g e t t i n g e x a c t l y 12 q u e s t i o n s c o r r e c t : < / h 3 > < p c l a s s = β² k a t e x β b l o c k β² > < s p a n c l a s s = " k a t e x β d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m i > P < / m i > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > X < / m i > < m o > = < / m o > < m n > 12 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m n > 125970 < / m n > < m o > Γ < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 0.2 < / m n > < m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m n > 12 < / m n > < / m s u p > < m o > Γ < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 0.8 < / m n > < m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m n > 8 < / m n > < / m s u p > < m o > = < / m o > < m n > 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > P ( X = 12 ) = 125970 Γ ( 0.2 ) 12 Γ ( 0.8 ) 8 = 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n β r i g h t : 0.13889 e m ; " > P < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n β r i g h t : 0.07847 e m ; " > X < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 12 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 125970 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 0.2 < / s p a n > < s p a n c l a s s = " m c l o s e " > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > Γ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v e r t i c a l β a l i g n : β 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 0.8 < / s p a n > < s p a n c l a s s = " m c l o s e " > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : β 3.113 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 8 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
</span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> is the number of trials, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> is the number of successes, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> is the probability of success, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\binom{n}{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> is the binomial coefficient.</p>
<h3>Q: What is the binomial coefficient <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mn>20</mn><mn>12</mn></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\binom{20}{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2451em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8951em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">20</span></span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span>?</h3>
<h3>A: The binomial coefficient <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mn>20</mn><mn>12</mn></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\binom{20}{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2451em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8951em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">20</span></span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> can be calculated using the formula:</h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><mrow><mi>n</mi><mo stretchy="false">!</mo></mrow><mrow><mi>k</mi><mo stretchy="false">!</mo><mo stretchy="false">(</mo><mi>n</mi><mo>β</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">!</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\binom{n}{k} = \frac{n!}{k!(n-k)!}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3074em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">!</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mclose">!</span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo stretchy="false">!</mo></mrow><annotation encoding="application/x-tex">n!</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">n</span><span class="mclose">!</span></span></span></span> is the factorial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>. Plugging in the values, we get:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mn>20</mn><mn>12</mn></mfrac><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><mrow><mn>20</mn><mo stretchy="false">!</mo></mrow><mrow><mn>12</mn><mo stretchy="false">!</mo><mo stretchy="false">(</mo><mn>20</mn><mo>β</mo><mn>12</mn><mo stretchy="false">)</mo><mo stretchy="false">!</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>20</mn><mo stretchy="false">!</mo></mrow><mrow><mn>12</mn><mo stretchy="false">!</mo><mn>8</mn><mo stretchy="false">!</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\binom{20}{12} = \frac{20!}{12!(20-12)!} = \frac{20!}{12!8!}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">12</span></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">20</span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3074em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">12</span><span class="mclose">!</span><span class="mopen">(</span><span class="mord">20</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">12</span><span class="mclose">)!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">20</span><span class="mclose">!</span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">12</span><span class="mclose">!</span><span class="mord">8</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">20</span><span class="mclose">!</span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<h3>Q: How do we evaluate the factorials?</h3>
<h3>A: To evaluate the factorials, we can use the following formula:</h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>n</mi><mo stretchy="false">!</mo><mo>=</mo><mi>n</mi><mo>Γ</mo><mo stretchy="false">(</mo><mi>n</mi><mo>β</mo><mn>1</mn><mo stretchy="false">)</mo><mo>Γ</mo><mo stretchy="false">(</mo><mi>n</mi><mo>β</mo><mn>2</mn><mo stretchy="false">)</mo><mo>Γ</mo><mo>β¦</mo><mo>Γ</mo><mn>2</mn><mo>Γ</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">n</span><span class="mclose">!</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">β¦</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p>
<p>Plugging in the values, we get:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>20</mn><mo stretchy="false">!</mo><mo>=</mo><mn>20</mn><mo>Γ</mo><mn>19</mn><mo>Γ</mo><mn>18</mn><mo>Γ</mo><mo>β¦</mo><mo>Γ</mo><mn>2</mn><mo>Γ</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">20! = 20 \times 19 \times 18 \times \ldots \times 2 \times 1
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">20</span><span class="mclose">!</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">20</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">19</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">β¦</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>12</mn><mo stretchy="false">!</mo><mo>=</mo><mn>12</mn><mo>Γ</mo><mn>11</mn><mo>Γ</mo><mn>10</mn><mo>Γ</mo><mo>β¦</mo><mo>Γ</mo><mn>2</mn><mo>Γ</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">12! = 12 \times 11 \times 10 \times \ldots \times 2 \times 1
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">12</span><span class="mclose">!</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">11</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">β¦</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>8</mn><mo stretchy="false">!</mo><mo>=</mo><mn>8</mn><mo>Γ</mo><mn>7</mn><mo>Γ</mo><mn>6</mn><mo>Γ</mo><mo>β¦</mo><mo>Γ</mo><mn>2</mn><mo>Γ</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">8! = 8 \times 7 \times 6 \times \ldots \times 2 \times 1
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">8</span><span class="mclose">!</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">7</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">β¦</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p>
<h3>Q: What is the value of the binomial coefficient <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mn>20</mn><mn>12</mn></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\binom{20}{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2451em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8951em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">20</span></span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span>?</h3>
<h3>A: Now that we have evaluated the factorials, we can calculate the binomial coefficient:</h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mn>20</mn><mn>12</mn></mfrac><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><mrow><mn>20</mn><mo stretchy="false">!</mo></mrow><mrow><mn>12</mn><mo stretchy="false">!</mo><mn>8</mn><mo stretchy="false">!</mo></mrow></mfrac><mo>=</mo><mfrac><mn>2432902008176640000</mn><mn>479001600</mn></mfrac><mo>=</mo><mn>125970</mn></mrow><annotation encoding="application/x-tex">\binom{20}{12} = \frac{20!}{12!8!} = \frac{2432902008176640000}{479001600} = 125970
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">12</span></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">20</span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">12</span><span class="mclose">!</span><span class="mord">8</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">20</span><span class="mclose">!</span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">479001600</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2432902008176640000</span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">125970</span></span></span></span></span></p>
<h3>Q: How do we calculate the probability of getting exactly 12 questions correct?</h3>
<h3>A: Now that we have calculated the binomial coefficient, we can calculate the probability of getting exactly 12 questions correct:</h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>12</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mn>20</mn><mn>12</mn></mfrac><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo><mn>0.2</mn><msup><mo stretchy="false">)</mo><mn>12</mn></msup><mo stretchy="false">(</mo><mn>0.8</mn><msup><mo stretchy="false">)</mo><mn>8</mn></msup><mo>=</mo><mn>125970</mn><mo>Γ</mo><mo stretchy="false">(</mo><mn>0.2</mn><msup><mo stretchy="false">)</mo><mn>12</mn></msup><mo>Γ</mo><mo stretchy="false">(</mo><mn>0.8</mn><msup><mo stretchy="false">)</mo><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">P(X = 12) = \binom{20}{12} (0.2)^{12} (0.8)^{8} = 125970 \times (0.2)^{12} \times (0.8)^{8}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">12</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">12</span></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">20</span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mopen">(</span><span class="mord">0.2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0.8</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">125970</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0.2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0.8</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<h3>Q: What is the value of the probability of getting exactly 12 questions correct?</h3>
<h3>A: Now that we have evaluated the exponents, we can calculate the probability of getting exactly 12 questions correct:</h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>12</mn><mo stretchy="false">)</mo><mo>=</mo><mn>125970</mn><mo>Γ</mo><mo stretchy="false">(</mo><mn>0.2</mn><msup><mo stretchy="false">)</mo><mn>12</mn></msup><mo>Γ</mo><mo stretchy="false">(</mo><mn>0.8</mn><msup><mo stretchy="false">)</mo><mn>8</mn></msup><mo>=</mo><mn>0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000</mn></mrow><annotation encoding="application/x-tex">P(X = 12) = 125970 \times (0.2)^{12} \times (0.8)^{8} = 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">12</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">125970</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0.2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0.8</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000</span></span></span></span></span></p>
P ( X = k ) = ( k n β ) p k ( 1 β p ) n β k < / s p an >< / p >< p > w h ere < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > n < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< / s p an >< / s p an >< / s p an > i s t h e n u mb ero f t r ia l s , < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > k < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > k < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in β r i g h t : 0.03148 e m ; " > k < / s p an >< / s p an >< / s p an >< / s p an > i s t h e n u mb ero f s u ccesses , < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > p < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > p < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.625 e m ; v er t i c a l β a l i g n : β 0.1944 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > p < / s p an >< / s p an >< / s p an >< / s p an > i s t h e p ro babi l i t yo f s u ccess , an d < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c l in e t hi c kn ess = "0 p x " >< mi > n < / mi >< mi > k < / mi >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > ( k n β ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.2 e m ; v er t i c a l β a l i g n : β 0.35 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ( < / s p an >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7454 e m ; " >< s p an s t y l e = " t o p : β 2.355 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in β r i g h t : 0.03148 e m ; " > k < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.144 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > i s t h e bin o mia l coe ff i c i e n t . < / p >< h 3 > Q : Wha t i s t h e bin o mia l coe ff i c i e n t < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c l in e t hi c kn ess = "0 p x " >< mn > 20 < / mn >< mn > 12 < / mn >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > ( 12 20 β ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.2451 e m ; v er t i c a l β a l i g n : β 0.35 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ( < / s p an >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8951 e m ; " >< s p an s t y l e = " t o p : β 2.355 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.144 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 20 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > ? < / h 3 >< h 3 > A : T h e bin o mia l coe ff i c i e n t < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c l in e t hi c kn ess = "0 p x " >< mn > 20 < / mn >< mn > 12 < / mn >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > ( 12 20 β ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.2451 e m ; v er t i c a l β a l i g n : β 0.35 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ( < / s p an >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8951 e m ; " >< s p an s t y l e = " t o p : β 2.355 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.144 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 20 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > c anb ec a l c u l a t e d u s in g t h e f or m u l a :< / h 3 >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c l in e t hi c kn ess = "0 p x " >< mi > n < / mi >< mi > k < / mi >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< m o >=< / m o >< m f r a c >< m ro w >< mi > n < / mi >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< m ro w >< mi > k < / mi >< m os t re t c h y = " f a l se " > ! < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mi > n < / mi >< m o > β < / m o >< mi > k < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > ( k n β ) = k ! ( n β k )! n ! β < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v er t i c a l β a l i g n : β 0.95 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ( < / s p an >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.1076 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in β r i g h t : 0.03148 e m ; " > k < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ) < / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.3074 e m ; v er t i c a l β a l i g n : β 0.936 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in β r i g h t : 0.03148 e m ; " > k < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > β < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in β r i g h t : 0.03148 e m ; " > k < / s p an >< s p an c l a ss = " m c l ose " > )! < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c β l in e " s t y l e = " b or d er β b o tt o m β w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.936 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p > w h ere < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > n < / mi >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > n ! < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< / s p an >< / s p an >< / s p an > i s t h e f a c t or ia l o f < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > n < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< / s p an >< / s p an >< / s p an > . Pl ugg in g in t h e v a l u es , w e g e t :< / p >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c l in e t hi c kn ess = "0 p x " >< mn > 20 < / mn >< mn > 12 < / mn >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< m o >=< / m o >< m f r a c >< m ro w >< mn > 20 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< m ro w >< mn > 12 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 20 < / mn >< m o > β < / m o >< mn > 12 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< / m f r a c >< m o >=< / m o >< m f r a c >< m ro w >< mn > 20 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< m ro w >< mn > 12 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< mn > 8 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > ( 12 20 β ) = 12 ! ( 20 β 12 )! 20 ! β = 12 ! 8 ! 20 ! β < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v er t i c a l β a l i g n : β 0.95 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ( < / s p an >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 12 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 20 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ) < / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.3074 e m ; v er t i c a l β a l i g n : β 0.936 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 12 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 20 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > β < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 12 < / s p an >< s p an c l a ss = " m c l ose " > )! < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c β l in e " s t y l e = " b or d er β b o tt o m β w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 20 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.936 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0574 e m ; v er t i c a l β a l i g n : β 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 12 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< s p an c l a ss = " m or d " > 8 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c β l in e " s t y l e = " b or d er β b o tt o m β w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 20 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 3 > Q : Ho w d o w ee v a l u a t e t h e f a c t or ia l s ? < / h 3 >< h 3 > A : T oe v a l u a t e t h e f a c t or ia l s , w ec an u se t h e f o ll o w in g f or m u l a :< / h 3 >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mi > n < / mi >< m os t re t c h y = " f a l se " > ! < / m o >< m o >=< / m o >< mi > n < / mi >< m o > Γ < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mi > n < / mi >< m o > β < / m o >< mn > 1 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m o > Γ < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mi > n < / mi >< m o > β < / m o >< mn > 2 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m o > Γ < / m o >< m o > β¦ < / m o >< m o > Γ < / m o >< mn > 2 < / mn >< m o > Γ < / m o >< mn > 1 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > n ! = n Γ ( n β 1 ) Γ ( n β 2 ) Γ β¦ Γ 2 Γ 1 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > β < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > β < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " minn er " > β¦ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p > Pl ugg in g in t h e v a l u es , w e g e t :< / p >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mn > 20 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< m o >=< / m o >< mn > 20 < / mn >< m o > Γ < / m o >< mn > 19 < / mn >< m o > Γ < / m o >< mn > 18 < / mn >< m o > Γ < / m o >< m o > β¦ < / m o >< m o > Γ < / m o >< mn > 2 < / mn >< m o > Γ < / m o >< mn > 1 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > 20 ! = 20 Γ 19 Γ 18 Γ β¦ Γ 2 Γ 1 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 20 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 20 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 19 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 18 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " minn er " > β¦ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mn > 12 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< m o >=< / m o >< mn > 12 < / mn >< m o > Γ < / m o >< mn > 11 < / mn >< m o > Γ < / m o >< mn > 10 < / mn >< m o > Γ < / m o >< m o > β¦ < / m o >< m o > Γ < / m o >< mn > 2 < / mn >< m o > Γ < / m o >< mn > 1 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > 12 ! = 12 Γ 11 Γ 10 Γ β¦ Γ 2 Γ 1 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 12 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 12 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 11 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 10 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " minn er " > β¦ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mn > 8 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< m o >=< / m o >< mn > 8 < / mn >< m o > Γ < / m o >< mn > 7 < / mn >< m o > Γ < / m o >< mn > 6 < / mn >< m o > Γ < / m o >< m o > β¦ < / m o >< m o > Γ < / m o >< mn > 2 < / mn >< m o > Γ < / m o >< mn > 1 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > 8 ! = 8 Γ 7 Γ 6 Γ β¦ Γ 2 Γ 1 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6944 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 8 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 8 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 7 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 6 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6667 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " minn er " > β¦ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 3 > Q : Wha t i s t h e v a l u eo f t h e bin o mia l coe ff i c i e n t < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c l in e t hi c kn ess = "0 p x " >< mn > 20 < / mn >< mn > 12 < / mn >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > ( 12 20 β ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.2451 e m ; v er t i c a l β a l i g n : β 0.35 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ( < / s p an >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8951 e m ; " >< s p an s t y l e = " t o p : β 2.355 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.144 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 20 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > ? < / h 3 >< h 3 > A : N o wt ha tw e ha v ee v a l u a t e d t h e f a c t or ia l s , w ec an c a l c u l a t e t h e bin o mia l coe ff i c i e n t :< / h 3 >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c l in e t hi c kn ess = "0 p x " >< mn > 20 < / mn >< mn > 12 < / mn >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< m o >=< / m o >< m f r a c >< m ro w >< mn > 20 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< m ro w >< mn > 12 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< mn > 8 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< / m f r a c >< m o >=< / m o >< m f r a c >< mn > 2432902008176640000 < / mn >< mn > 479001600 < / mn >< / m f r a c >< m o >=< / m o >< mn > 125970 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > ( 12 20 β ) = 12 ! 8 ! 20 ! β = 479001600 2432902008176640000 β = 125970 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v er t i c a l β a l i g n : β 0.95 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ( < / s p an >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 12 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 20 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ) < / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0574 e m ; v er t i c a l β a l i g n : β 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 12 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< s p an c l a ss = " m or d " > 8 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c β l in e " s t y l e = " b or d er β b o tt o m β w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 20 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v er t i c a l β a l i g n : β 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 479001600 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c β l in e " s t y l e = " b or d er β b o tt o m β w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2432902008176640000 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 125970 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 3 > Q : Ho w d o w ec a l c u l a t e t h e p ro babi l i t yo f g e tt in g e x a c tl y 12 q u es t i o n scorrec t ? < / h 3 >< h 3 > A : N o wt ha tw e ha v ec a l c u l a t e d t h e bin o mia l coe ff i c i e n t , w ec an c a l c u l a t e t h e p ro babi l i t yo f g e tt in g e x a c tl y 12 q u es t i o n scorrec t :< / h 3 >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mi > P < / mi >< m os t re t c h y = " f a l se " > ( < / m o >< mi > X < / mi >< m o >=< / m o >< mn > 12 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c l in e t hi c kn ess = "0 p x " >< mn > 20 < / mn >< mn > 12 < / mn >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 0.2 < / mn >< m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< mn > 12 < / mn >< / m s u p >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 0.8 < / mn >< m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< mn > 8 < / mn >< / m s u p >< m o >=< / m o >< mn > 125970 < / mn >< m o > Γ < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 0.2 < / mn >< m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< mn > 12 < / mn >< / m s u p >< m o > Γ < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 0.8 < / mn >< m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< mn > 8 < / mn >< / m s u p >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > P ( X = 12 ) = ( 12 20 β ) ( 0.2 ) 12 ( 0.8 ) 8 = 125970 Γ ( 0.2 ) 12 Γ ( 0.8 ) 8 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in β r i g h t : 0.13889 e m ; " > P < / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in β r i g h t : 0.07847 e m ; " > X < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 12 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v er t i c a l β a l i g n : β 0.95 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ( < / s p an >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : β 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 12 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 20 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ) < / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 0.2 < / s p an >< s p an c l a ss = " m c l ose " >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 0.8 < / s p an >< s p an c l a ss = " m c l ose " >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 8 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 125970 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 0.2 < / s p an >< s p an c l a ss = " m c l ose " >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 0.8 < / s p an >< s p an c l a ss = " m c l ose " >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 8 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 3 > Q : Wha t i s t h e v a l u eo f t h e p ro babi l i t yo f g e tt in g e x a c tl y 12 q u es t i o n scorrec t ? < / h 3 >< h 3 > A : N o wt ha tw e ha v ee v a l u a t e d t h ee x p o n e n t s , w ec an c a l c u l a t e t h e p ro babi l i t yo f g e tt in g e x a c tl y 12 q u es t i o n scorrec t :< / h 3 >< p c l a ss = β² ka t e x β b l oc k β² >< s p an c l a ss = " ka t e x β d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mi > P < / mi >< m os t re t c h y = " f a l se " > ( < / m o >< mi > X < / mi >< m o >=< / m o >< mn > 12 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< mn > 125970 < / mn >< m o > Γ < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 0.2 < / mn >< m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< mn > 12 < / mn >< / m s u p >< m o > Γ < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 0.8 < / mn >< m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< mn > 8 < / mn >< / m s u p >< m o >=< / m o >< mn > 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > P ( X = 12 ) = 125970 Γ ( 0.2 ) 12 Γ ( 0.8 ) 8 = 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< / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in β r i g h t : 0.13889 e m ; " > P < / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in β r i g h t : 0.07847 e m ; " > X < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 12 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 125970 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 0.2 < / s p an >< s p an c l a ss = " m c l ose " >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > Γ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.1141 e m ; v er t i c a l β a l i g n : β 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 0.8 < / s p an >< s p an c l a ss = " m c l ose " >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t " >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : β 3.113 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 8 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 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