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🔀 Permutations & Combinations Lab

Same numbers n and r —
the answer changes based on whether order matters

Choosing r people out of n, if you care about "who's 1st, who's 2nd," that's a permutation; if you only care about "who got picked," that's a combination. Tap the tabs and compare directly how different the two counts really are.

A permutation (P) is a count where order matters. Like "how many ways to decide 1st, 2nd, and 3rd place," who ends up in which spot matters. A combination (C) is a count where order doesn't matter. Like "how many ways to choose 3 representatives," only who got picked matters, not the order they were picked in.

So a combination is always less than or equal to the matching permutation. Even choosing the same people, if you don't distinguish the order, several permutations collapse into a single combination. They overlap by exactly the number of ways to arrange r people in order (r!).

This difference matters in everyday life too. Deciding places in a foot race is a permutation, because order matters. But choosing 3 class representatives is a combination, because only who got picked matters, not the order. A case like a lock's passcode, where a different order gives a completely different result, is also calculated as a permutation.

Total people (n)5
People chosen (r)2
₅P₂ = 5 × 4
20 ways