#### How many ways can you arrange 3 letters with 1 repeat?

Compare the permutations of the letters A,B,C with those of the same number of letters, 3, but with one repeated letter $$ \rightarrow $$ A, A, B

**All the different arrangements of the letters A, B, C**

**All the different arrangements of the letters A, A, B**

$$ \frac{ 3!}{2!} = \frac{(3 \cdot 2 \cdot 1)}{(2 \cdot 1)} = 3 $$

If A out of N items are identical, then the number of different permutations of the N items is $$ \frac{ N! }{ A! } $$

If a set of N items contains A identical items, B identical items, and C identical items etc.., then the total number of different permutations of N objects is $ \frac{ N! }{ A! \cdot B! \cdot C! ... ! } $

### Permutation **Practice Problems**

There is only one letter that repeats.

Since the letter *o* appears twice we need to divide by 2!

The letters a and p are the ones that repeat.

Since the letter *a* occurs twice and the letter *p* also occurs twice, we have to divide by 2! two times.

The repeats: there are four occurrences of the letter *i*, four occurrences of the letter *s* , and two occurrences of the letter *p*

The total number of letters is 11.