The Hawaiian Alphabet Has 12 Letters. How Many Permutations Are Possible For Five Of These Letters? N P R = N ! ( N − R ) ! {}_n P_r = \frac{n!}{(n-r)!} N ​ P R ​ = ( N − R )! N ! ​

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The Hawaiian Alphabet and Permutations: Unlocking the Power of Combinations

The Hawaiian alphabet, also known as the Hawaiian language alphabet, consists of 12 letters. These letters are used to write the Hawaiian language, which is an official language of the state of Hawaii in the United States. In this article, we will explore the concept of permutations and how many possible combinations can be formed using five of these letters.

Permutations are a fundamental concept in mathematics that deals with the arrangement of objects in a specific order. In other words, permutations refer to the number of ways in which objects can be arranged or ordered. For example, if we have three letters A, B, and C, the possible permutations are ABC, ACB, BAC, BCA, CAB, and CBA.

The formula for permutations is given by:

nPr=n!(n−r)!{}_n P_r = \frac{n!}{(n-r)!}

where:

  • nPr{}_n P_r is the number of permutations of n objects taken r at a time
  • n is the total number of objects
  • r is the number of objects being chosen
  • ! denotes the factorial of a number, which is the product of all positive integers less than or equal to that number

In this case, we have 12 letters in the Hawaiian alphabet, and we want to find the number of permutations possible for five of these letters. Using the formula above, we can calculate the number of permutations as follows:

12P5=12!(12−5)!{}_{12} P_5 = \frac{12!}{(12-5)!}

To calculate the factorials, we need to multiply all positive integers less than or equal to the given number. For example, the factorial of 12 (12!) is:

12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

Similarly, the factorial of (12-5) or 7 is:

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Now that we have calculated the factorials, we can substitute the values into the formula:

12P5=12!7!{}_{12} P_5 = \frac{12!}{7!}

To simplify the expression, we can cancel out the common factors in the numerator and denominator:

12P5=12×11×10×9×8×7!7!{}_{12} P_5 = \frac{12 × 11 × 10 × 9 × 8 × 7!}{7!}

The 7! in the numerator and denominator cancels out, leaving us with:

12P5=12×11×10×9×8{}_{12} P_5 = 12 × 11 × 10 × 9 × 8

Now that we have simplified the expression, we can calculate the final answer:

12P5=12×11×10×9×8{}_{12} P_5 = 12 × 11 × 10 × 9 × 8

= 95,040

Therefore, the number of permutations possible for five letters in the Hawaiian alphabet is 95,040.

In this article, we explored the concept of permutations and how many possible combinations can be formed using five letters in the Hawaiian alphabet. We used the formula for permutations to calculate the number of permutations and simplified the expression to arrive at the final answer. The result shows that there are 95,040 possible permutations for five letters in the Hawaiian alphabet.

  • Q: What is the Hawaiian alphabet? A: The Hawaiian alphabet consists of 12 letters.
  • Q: What is the formula for permutations? A: The formula for permutations is given by nPr=n!(n−r)!{}_n P_r = \frac{n!}{(n-r)!}.
  • Q: How many permutations are possible for five letters in the Hawaiian alphabet? A: The number of permutations possible for five letters in the Hawaiian alphabet is 95,040.
  • "Hawaiian Language Alphabet" by the University of Hawaii at Manoa
  • "Permutations" by Math Is Fun
  • "Factorial" by Wolfram MathWorld
    The Hawaiian Alphabet and Permutations: A Q&A Guide

In our previous article, we explored the concept of permutations and how many possible combinations can be formed using five letters in the Hawaiian alphabet. We used the formula for permutations to calculate the number of permutations and simplified the expression to arrive at the final answer. In this article, we will provide a Q&A guide to help you better understand the concept of permutations and the Hawaiian alphabet.

Q: What is the Hawaiian alphabet? A: The Hawaiian alphabet consists of 12 letters: A, E, I, O, U, H, K, L, M, N, P, and W.

Q: What is the formula for permutations? A: The formula for permutations is given by:

nPr=n!(n−r)!{}_n P_r = \frac{n!}{(n-r)!}

where:

  • nPr{}_n P_r is the number of permutations of n objects taken r at a time
  • n is the total number of objects
  • r is the number of objects being chosen
  • ! denotes the factorial of a number, which is the product of all positive integers less than or equal to that number

Q: How many permutations are possible for five letters in the Hawaiian alphabet? A: The number of permutations possible for five letters in the Hawaiian alphabet is 95,040.

Q: What is the difference between permutations and combinations? A: Permutations refer to the number of ways in which objects can be arranged or ordered, while combinations refer to the number of ways in which objects can be chosen without regard to order.

Q: How do I calculate the factorial of a number? A: To calculate the factorial of a number, you need to multiply all positive integers less than or equal to that number. For example, the factorial of 5 (5!) is:

5! = 5 × 4 × 3 × 2 × 1 = 120

Q: What is the significance of the Hawaiian alphabet? A: The Hawaiian alphabet is an important part of Hawaiian culture and language. It is used to write the Hawaiian language, which is an official language of the state of Hawaii in the United States.

Q: Can I use the formula for permutations to calculate the number of permutations for more than five letters? A: Yes, you can use the formula for permutations to calculate the number of permutations for more than five letters. Simply substitute the values of n and r into the formula and calculate the result.

Q: What is the relationship between permutations and probability? A: Permutations are related to probability in that they can be used to calculate the probability of certain events occurring. For example, if you want to calculate the probability of drawing a certain card from a deck of cards, you can use permutations to calculate the number of possible outcomes.

Q: Can I use the formula for permutations to calculate the number of permutations for a set of objects with repeating elements? A: Yes, you can use the formula for permutations to calculate the number of permutations for a set of objects with repeating elements. However, you need to take into account the fact that the objects are repeating, and adjust the formula accordingly.

In this article, we provided a Q&A guide to help you better understand the concept of permutations and the Hawaiian alphabet. We covered a range of topics, from the formula for permutations to the significance of the Hawaiian alphabet. We hope that this guide has been helpful in clarifying any questions you may have had about permutations and the Hawaiian alphabet.

  • Q: What is the Hawaiian alphabet? A: The Hawaiian alphabet consists of 12 letters.
  • Q: What is the formula for permutations? A: The formula for permutations is given by nPr=n!(n−r)!{}_n P_r = \frac{n!}{(n-r)!}.
  • Q: How many permutations are possible for five letters in the Hawaiian alphabet? A: The number of permutations possible for five letters in the Hawaiian alphabet is 95,040.
  • "Hawaiian Language Alphabet" by the University of Hawaii at Manoa
  • "Permutations" by Math Is Fun
  • "Factorial" by Wolfram MathWorld