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

This is a multiset permutation because all 5 letters are arranged, order matters, and the two Ps are identical before arranging. Think of each letter as a distinct symbol or token, how can we arrange them?

Exact result

60

For item-group counts 2, 1, 1, 1, the number of distinct arrangements is 60.

APPLE has counts 2,1,1,1 because P appears twice while A, L, and E appear once. Start from 5! and divide by 2! to remove duplicate swaps of the repeated P.

Worked steps

Show your work

  1. Inputs: item-group counts = 2, 1, 1, 1
  2. Formula: T! / (n1! n2! n3! ...)
  3. Substitute: 5! / (2! 1! 1! 1!)
  4. Steps: start with 120 and divide by 2! = 2, 1! = 1, 1! = 1, 1! = 1
  5. Result: For item-group counts 2, 1, 1, 1, the number of distinct arrangements is 60.

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APPLE has counts 2,1,1,1 because P appears twice while A, L, and E appear once. Start from 5! and divide by 2! to remove duplicate swaps of the repeated P.

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