# Permutations with repeats

## Advanced Concepts

Repeat Items in Permutations

#### 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

$$\frac{( \text{total number of letters})!}{ ( \text{number of repeats})! }$$
$$\frac{ 3!}{2!} = \frac{(3 \cdot 2 \cdot 1)}{(2 \cdot 1)} = 3$$

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.

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