Permutation and Combination Calculator
Compute permutations and combinations for n and r with or without order.
Permutations and combinations
Permutations count ordered arrangements, combinations count selections without order.
Permutation (nPr):
Combination (nCr):
Formulas:
Permutations vs. combinations
Permutations (nPr) count arrangements where order matters — "ABC" and "CBA" are counted as different outcomes. Combinations (nCr) count selections where order doesn't matter — "ABC" and "CBA" are the same selection. Since every combination of r items can be arranged in r! different orders, nPr is always nCr multiplied by r! — which is exactly why the combination formula divides the permutation formula by an extra r!.
Frequently asked questions
How do I know whether to use permutation or combination? Ask whether order matters for your scenario: arranging runners into 1st/2nd/3rd place is a permutation (order matters), while choosing 3 people for a committee is a combination (order doesn't matter).
Why is the result "0" when r is greater than n? You can't select or arrange more items than exist in the group — for example, choosing 5 items from a group of only 3 is impossible, so both nPr and nCr correctly evaluate to zero.
Why does this use factorials? Factorial (n!) counts all possible full orderings of n items — permutations and combinations both build on this by "trimming" a full factorial down to only the r positions being arranged or selected.