Permutation and Combination Calculator

nPr (order matters) and nCr (order doesn't matter) for any n and r

Permutations (nPr) and combinations (nCr) both count ways of selecting r items from a set of n, but they answer genuinely different questions depending on whether order matters — permutations count arrangements where order is distinct, combinations count selections where it isn't, and using the wrong one for a given problem produces a systematically wrong count.

Why order matters for one but not the other

A permutation counts the number of ways to select and arrange r items from n, where a different order counts as a different outcome — selecting the same three items but in a different sequence is counted as a separate permutation. A combination counts the number of ways to select r items from n where order is irrelevant — the same three items, regardless of what order they were chosen in, count as a single combination. Because permutations count every distinct ordering separately, the permutation count for a given n and r is always larger than (or equal to) the combination count for the same n and r — specifically, larger by a factor equal to the number of ways those r items could themselves be arranged (r factorial).

A worked example

Choosing 3 people from a group for three distinct, ordered roles (like first, second, third place) uses permutations, since who fills which specific role matters — choosing the same 3 people simply as an unordered team or committee uses combinations, since which role each person "has" doesn't matter, only who's on the team. The same underlying selection of 3 people from the group produces a much larger permutation count than combination count, since the permutation count also accounts for all the ways those 3 could be assigned to the distinct roles.

How this connects to your other math and probability calculations

Permutation and combination counts are frequently building blocks for probability calculations — the Probability Calculator covers the broader probability calculations that permutation/combination counts often feed into as the "number of favorable outcomes" or "total possible outcomes" inputs.

Common mistakes

Using a combination formula when a problem actually requires distinguishing between different orderings (or vice versa) is the most common error — before applying either formula, explicitly ask whether swapping the order of two selected items would create a genuinely different outcome for the specific problem at hand; if yes, use permutations, if no, use combinations.