Calculate the number of combinations (nCr), the number of ways to choose r items from a set of n distinct items, where order doesn't matter.
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nCr = n! ÷ (r! × (n − r)!), the number of ways to choose r items from a set of n distinct items, where the order of selection doesn't matter. For example, choosing a 3-person committee from 10 people gives 10C3 = 120 possible committees, far fewer than the 720 permutations of the same items, since here each group of 3 people counts only once regardless of selection order.
Since combinations don't distinguish between different orderings of the same group, nCr is always nPr divided by r! (the number of ways to reorder r items). This is exactly why choosing a committee (120 ways) produces a smaller count than arranging medal positions (720 ways) from the same 10 people and group size of 3.
Use combinations whenever the order of selection doesn't create a meaningfully different outcome, like choosing lottery numbers, forming a team, or picking a subset of items, versus permutations for cases like rankings, seating arrangements, or passwords where order matters.
'n choose r', written nCr or sometimes with a binomial coefficient symbol, is the standard shorthand for this combination formula, and is also the coefficient used in binomial expansions and binomial probability.
nCr with r=0 always equals 1, there's exactly one way to choose nothing from any set (the empty selection).
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