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

Permutation without Repeats

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

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

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

Problem 1

How many ways can you arrange the letters of the word 'loose'?

There is only one letter that repeats.

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

Permutation of Loose
Problem 2

How many can you arrange the letters of the word ' appearing'? stuck

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.

Problem 3

This question revolves around a permutation of a word with many repeated letters.

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.

permutations of mississippi repeats
to Permutations

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