The Hawaiian alphabet has 12 letters. How many permutations are possible for five of these letters? $$ { }_{n} P_{r}=\frac{n!}{(n-r)!} $$

Question

UpStudy AI · Accepted Answer

The permutation formula is given by

$$
{}_{n}P_{r} = \frac{n!}{(n-r)!}
$$

Here, $$ n = 12 $$ and $$ r = 5 $$. Substituting these values, we get

$$
{}_{12}P_{5} = \frac{12!}{(12-5)!} = \frac{12!}{7!}
$$

This expression can be simplified by writing out the factorials:

$$
\frac{12!}{7!} = 12 \times 11 \times 10 \times 9 \times 8
$$

Now, calculate the product step by step:

1. $$ 12 \times 11 = 132 $$
2. $$ 132 \times 10 = 1320 $$
3. $$ 1320 \times 9 = 11880 $$
4. $$ 11880 \times 8 = 95040 $$

Thus, the number of permutations is

$$
\boxed{95040}
$$
There are 95,040 possible permutations of five letters from the Hawaiian alphabet.

The Hawaiian alphabet has 12 letters. How many permutations are possible for five of these letters? \[ { }_{n} P_{r}=\frac{n!}{(n-r)!} \]