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Permutations

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Permutations

A permutation counts the number of ways to choose and arrange r items from a set of n, where order matters. Learn the formula nPr = n! divided by (n minus r) factorial, work through an example arranging letters, and see how permutations differ from combinations, where order doesn't matter.

What a permutation is

A permutation counts the number of ways to choose and arrange r items from a set of n items, where order matters. Arranging A then B is counted separately from arranging B then A, since they form a different sequence.

The permutations formula nPr equals n factorial divided by (n minus r) factorial. Worked example: choosing and arranging 2 letters from 4 (A, B, C, D) gives 4P2 = 4!/(4-2)! = 24/2 = 12 permutations. P(n,r) = n! (n − r)! 4P2 = 4! / (4 − 2)! = 24 / 2 4P2 = 12 ways to arrange 2 of 4
The permutation formula, with 4P2 = 12: the number of ways to arrange 2 of 4 letters in order.

The permutation formula

The number of permutations of r items chosen from n is P(n,r) = n! / (n − r)!, where n! is factorial notation. Dividing by (n − r)! removes the arrangements of the items you didn't choose, leaving only the ordered arrangements of the r you did.

Worked example

How many ways can you choose and order 2 letters from {A, B, C, D}? Using the formula: 4P2 = 4! / (4 − 2)! = 24 / 2 = 12. Listing them confirms it: AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, DC — twelve ordered pairs.

Permutations vs. combinations

The key question is always: does order matter? If yes, it's a permutation. If the order of selection doesn't matter — choosing a group rather than arranging a sequence — it's a combination instead, which uses a related but different formula.

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