Why do we divide by the factorials of the counts of repeating letters when calculating permutations?

Conditions

• Calculating permutations of a multiset.
• Some elements in the set are indistinguishable.

Reasoning, step by step

1. Start with the assumption that all $n$ items are distinct, yielding $n!$ arrangements.
2. Recognize that identical items (e.g., three 'A's) are interchangeable without changing the outcome.
3. Calculate the number of internal swaps for the identical group, which is $k!$.
4. Divide the total $n!$ by $k!$ to remove the redundant counts generated by treating identical items as distinct.
5. Repeat this division for every group of identical items.

Example

The script explains: 'Swapping identical letters does not create a new arrangement. To correct the overcounting from the $7!$ calculation, we divide by the factorial of the counts of each repeating letter group.'

Common misconceptions

• Believing that dividing by the count itself (e.g., 3) is sufficient instead of its factorial (3!).
• Thinking that identical items still contribute to uniqueness based on their original position.

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Connected concepts

Explore next

• What happens if there are multiple groups of different repeated letters?
• How does this principle apply to combinations rather than permutations?

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