Permutation & Combination Calculator
Calculate the number of permutations (order matters) or combinations (order doesn't matter) for choosing r items from n.
Permutations vs. combinations
The key difference is whether order matters. Arranging 3 people in a race (1st, 2nd, 3rd place) is a permutation, since who comes in 1st versus 2nd matters. Choosing 3 people for a committee is a combination, since the group is the same regardless of the order they were selected in.
Because permutations count every possible ordering separately, nPr is always greater than or equal to nCr for the same n and r — combinations essentially group together all the orderings that contain the same items.
A worked example
Choosing 3 winners (1st, 2nd, 3rd prize) from 10 entrants is a permutation: 10P3 = 10!/(10-3)! = 720 possible outcomes, since each different order counts separately.
Choosing 3 people from 10 to form a team (no ranking) is a combination: 10C3 = 10!/(3!×7!) = 120 possible outcomes — much smaller, since it doesn't matter which order the 3 were chosen in.
Frequently asked questions
How do I know whether to use permutation or combination?
Ask whether order matters for your situation. If rearranging the same items creates a different outcome (like race placements), use permutation. If the same items in any order count as the same result (like a committee), use combination.
Why is nPr always bigger than nCr for the same n and r?
Because permutations count every possible ordering of the same r items as separate outcomes, while combinations group all those orderings together as one outcome — so nPr is always a multiple of nCr (specifically, nPr = nCr × r!).