A Magician Performing A Magic Act Has Asked For 2 Volunteers From The Audience To Help With His Routine. If There Are 250 People In The Audience, Which Of The Following Expressions Represents The Number Of Possible Pairs Of Volunteers?A.
A Magical Math Problem: Finding the Number of Possible Pairs of Volunteers
In this problem, we are tasked with finding the number of possible pairs of volunteers that a magician can choose from an audience of 250 people. This problem involves combinations, a fundamental concept in mathematics that deals with selecting items from a larger set without regard to the order of selection.
Combinations are used to calculate the number of ways to choose a certain number of items from a larger set, without regard to the order of selection. In this case, we want to choose 2 volunteers from an audience of 250 people. The order in which we choose the volunteers does not matter, as long as we have 2 volunteers.
The Formula for Combinations
The formula for combinations is given by:
C(n, k) = n! / (k!(n-k)!)
where n is the total number of items in the set, k is the number of items to choose, and ! denotes the factorial function.
Applying the Formula to Our Problem
In our problem, we have n = 250 (the total number of people in the audience) and k = 2 (the number of volunteers to choose). Plugging these values into the formula, we get:
C(250, 2) = 250! / (2!(250-2)!)
Simplifying the Expression
To simplify the expression, we can use the fact that 250! = 250 × 249 × 248 × ... × 3 × 2 × 1. We can also use the fact that 2! = 2 × 1 and (250-2)! = 248!. Substituting these values into the expression, we get:
C(250, 2) = (250 × 249 × 248 × ... × 3 × 2 × 1) / ((2 × 1) × (248 × 247 × ... × 3 × 2 × 1))
Cancelling Out Common Factors
We can cancel out the common factors in the numerator and denominator, leaving us with:
C(250, 2) = (250 × 249) / 2
Evaluating the Expression
Evaluating the expression, we get:
C(250, 2) = 62450
Therefore, the number of possible pairs of volunteers that the magician can choose from an audience of 250 people is 62450.
Here are a few additional examples to illustrate the concept of combinations:
- If a group of 10 people wants to choose 3 people to form a committee, how many possible committees can be formed?
- If a basketball team has 12 players and wants to choose 5 players to start the game, how many possible starting lineups can be formed?
Solutions to Additional Examples
- C(10, 3) = 10! / (3!(10-3)!) = (10 × 9 × 8) / (3 × 2 × 1) = 120
- C(12, 5) = 12! / (5!(12-5)!) = (12 × 11 × 10 × 9 × 8) / (5 × 4 × 3 × 2 × 1) = 792
In conclusion, combinations are a powerful tool for calculating the number of ways to choose a certain number of items from a larger set, without regard to the order of selection. By using the formula for combinations, we can easily calculate the number of possible pairs of volunteers that a magician can choose from an audience of 250 people.
A Magical Math Problem: Finding the Number of Possible Pairs of Volunteers - Q&A
In our previous article, we explored the concept of combinations and how to use the formula for combinations to calculate the number of possible pairs of volunteers that a magician can choose from an audience of 250 people. In this article, we will answer some frequently asked questions related to this problem.
Q: What is the difference between combinations and permutations?
A: Combinations and permutations are both used to calculate the number of ways to choose items from a larger set, but they differ in the order of selection. Permutations take into account the order of selection, while combinations do not.
Q: How do I calculate the number of combinations when the number of items is very large?
A: When the number of items is very large, it can be difficult to calculate the number of combinations using the formula for combinations. In such cases, you can use the fact that C(n, k) = n! / (k!(n-k)!) = n(n-1)(n-2)...(n-k+1) / k!. This can be calculated using a calculator or a computer program.
Q: Can I use the formula for combinations to calculate the number of possible pairs of volunteers if the audience is not a whole number?
A: No, the formula for combinations requires the number of items to be a whole number. If the audience is not a whole number, you will need to round up or down to the nearest whole number.
Q: How do I calculate the number of combinations when the number of items is not a whole number?
A: If the number of items is not a whole number, you can use the fact that C(n, k) = n! / (k!(n-k)!) = n(n-1)(n-2)...(n-k+1) / k!. This can be calculated using a calculator or a computer program.
Q: Can I use the formula for combinations to calculate the number of possible pairs of volunteers if the number of volunteers is not 2?
A: Yes, you can use the formula for combinations to calculate the number of possible pairs of volunteers if the number of volunteers is not 2. Simply plug in the values of n and k into the formula.
Q: How do I calculate the number of combinations if I want to choose more than 2 volunteers?
A: To calculate the number of combinations if you want to choose more than 2 volunteers, you can use the formula for combinations with k > 2. For example, if you want to choose 3 volunteers from an audience of 250 people, you can use the formula C(250, 3) = 250! / (3!(250-3)!) = (250 × 249 × 248) / (3 × 2 × 1).
Q: Can I use the formula for combinations to calculate the number of possible pairs of volunteers if the audience is not a fixed size?
A: No, the formula for combinations requires the number of items to be a fixed size. If the audience is not a fixed size, you will need to use a different method to calculate the number of possible pairs of volunteers.
In conclusion, combinations are a powerful tool for calculating the number of ways to choose a certain number of items from a larger set, without regard to the order of selection. By using the formula for combinations, we can easily calculate the number of possible pairs of volunteers that a magician can choose from an audience of 250 people. We hope this Q&A article has helped to clarify any questions you may have had about this problem.
- Calculate the number of possible pairs of volunteers that a magician can choose from an audience of 200 people.
- Calculate the number of possible pairs of volunteers that a magician can choose from an audience of 300 people.
- Calculate the number of possible pairs of volunteers that a magician can choose from an audience of 400 people.
Solutions to Practice Problems
- C(200, 2) = 200! / (2!(200-2)!) = (200 × 199) / 2 = 19900
- C(300, 2) = 300! / (2!(300-2)!) = (300 × 299) / 2 = 44950
- C(400, 2) = 400! / (2!(400-2)!) = (400 × 399) / 2 = 79900