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Combinations

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Combinations

A combination counts the number of ways to choose r items from a set of n, where order does not matter. Learn the formula nCr = n! divided by r! times (n minus r) factorial, work through an example choosing letters, and see how combinations relate to permutations.

What a combination is

A combination counts the number of ways to choose r items from a set of n items, where order does not matter. Choosing A and B is the same group as choosing B and A — there is only one combination, not two.

The combinations formula nCr equals n factorial divided by (r factorial times (n minus r) factorial). Worked example: choosing 2 letters from 4 (A, B, C, D) gives 4C2 = 4!/(2! times 2!) = 24/4 = 6 combinations. C(n,r) = n! r! (n − r)! 4C2 = 4! / (2! × 2!) = 24 / 4 4C2 = 6 ways to choose 2 of 4
The combination formula, with 4C2 = 6: the number of ways to choose 2 of 4 letters, order irrelevant.

The combinations formula

The number of combinations of r items chosen from n is C(n,r) = n! / (r! × (n − r)!), built from factorial notation. This is exactly the permutation formula divided by r!, because a combination collapses every reordering of the same r items into a single group.

Worked example

How many ways can you choose 2 letters from {A, B, C, D}, where order doesn't matter? Using the formula: 4C2 = 4! / (2! × 2!) = 24 / 4 = 6. Listing them confirms it: {A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D} — six unordered pairs, half as many as the twelve permutations of the same set.

Combinations vs. permutations

The deciding question is always whether order matters. Choosing lottery numbers or a committee is a combination, since the group is what counts, not the order chosen. Arranging a sequence, like a race's finishing order, is a permutation instead.

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