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Permutations with repeats

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

Problem 1) How many ways can you arrange the letters of the word 'loose'?