To determine the number of permutations of five letters out of a total of 12 letters in the Hawaiian alphabet, we'll use the permutation formula as given by [tex]\({ }_n P_r = \frac{n!}{(n-r)!}\)[/tex], where [tex]\( n \)[/tex] is the total number of items and [tex]\( r \)[/tex] is the number of items to arrange.
In this case:
- [tex]\( n = 12 \)[/tex] (total letters)
- [tex]\( r = 5 \)[/tex] (letters to arrange)
Here's the step-by-step solution:
1. Calculate [tex]\( n! \)[/tex]:
We need to find the factorial of [tex]\( n \)[/tex], which is [tex]\( 12! \)[/tex].
[tex]\[
12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479,001,600
\][/tex]
2. Calculate [tex]\( (n-r)! \)[/tex]:
Next, we need to find the factorial of [tex]\( n-r \)[/tex], which is [tex]\( 12-5 = 7 \)[/tex]. So, we need [tex]\( 7! \)[/tex].
[tex]\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5,040
\][/tex]
3. Calculate the number of permutations [tex]\( { }_{12}P_{5} \)[/tex]:
Using the permutation formula:
[tex]\[
{ }_{12}P_{5} = \frac{12!}{(12-5)!} = \frac{479,001,600}{5,040} = 95,040
\][/tex]
Therefore, the number of permutations for arranging five letters out of the twelve in the Hawaiian alphabet is [tex]\( 95,040 \)[/tex].
