Combinations & Permutations Calculator
Count the ways to choose or arrange r items from a set of n (nCr and nPr).
Result
Combinations (nCr)
120
Permutations (nPr)
720
How it works
Permutations and combinations both count selections, and the only difference is whether order matters. Permutations count arrangements — nPr = n!/(n−r)! — while combinations count groups regardless of order, nCr = nPr ÷ r!. That division by r! is the whole distinction. Choosing 3 people from 10 for three named roles gives 720 permutations; choosing 3 for an unnamed committee gives 720 ÷ 6 = 120 combinations, because the same three people can be ordered in 3! = 6 ways that all count as one group. The test is simple: ask whether swapping two chosen items changes the outcome. A podium does (gold and silver differ), a team does not. Combinations also have a pleasing symmetry — choosing 3 from 10 has the same count as choosing 7, since picking who is in is the same act as picking who is out.
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Frequently asked questions
What is the difference?
Permutations care about order (1-2-3 ≠ 3-2-1); combinations do not.
When do I use combinations?
When you only choose a group — lottery numbers, teams, committees — and order is irrelevant.
How many combinations are there in a 5-from-49 lottery?
1,906,884. That is C(49,5) = 49×48×47×46×45 ÷ 120 — and order does not matter, which is why the count uses combinations rather than the far larger permutation figure.
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