Permutation Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
How many distinct arrangements of the letters in the word MISSISSIPPI are there?

Solution

  1. 1
    Count total letters: 1111. Count repeats: M =1= 1, I =4= 4, S =4= 4, P =2= 2.
  2. 2
    Use the formula for permutations with repetition: n!n1!⋅n2!⋅…\frac{n!}{n_1! \cdot n_2! \cdot \ldots}.
  3. 3
    11!1!⋅4!⋅4!⋅2!=399168001⋅24⋅24⋅2=399168001152=34650\frac{11!}{1! \cdot 4! \cdot 4! \cdot 2!} = \frac{39916800}{1 \cdot 24 \cdot 24 \cdot 2} = \frac{39916800}{1152} = 34650.

Answer

3465034650
When elements repeat, divide the total permutations by the factorial of each group of identical items to avoid counting identical arrangements multiple times.

About Permutation

A permutation is an ordered arrangement of objects — the number of ways to choose and order rr items from nn distinct items is P(n,r)=n!(n−r)!P(n,r) = \frac{n!}{(n-r)!}.

Learn more about Permutation →

More Permutation Examples

Example 1 easy

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Example 3 easy

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Example 4 medium

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Related Concepts

FactorialCombination