


Permutations with repeatsAdvanced Concepts
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
Permutation Practice Problems
$
\frac{ N! }{
A! \cdot B! \cdot C! ... !
}
$
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!. 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) How many can you arrange the letters in the word 'Mississippi'? stuck
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.
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Permutations 