Permutation and Combination Calculator. Enter n (total number of items) and r (items to select) to calculate both permutations (nPr) and combinations (nCr) at once. The calculator returns the number of ordered arrangements and unordered selections, along with the step-by-step factorial breakdown for each formula. Also try the AND Probability Calculator.
A permutation counts arrangements where order matters — choosing A then B is different from B then A. A combination counts selections where order does not matter — {A, B} and {B, A} are treated as the same group. Use permutations for rankings or sequences, and combinations for groups or subsets. See also our calculate Bayesian A/B Test Variant B — Chance to Beat Control.
Without repetitions: nPr = n! / (n−r)! and nCr = n! / (r! × (n−r)!). With repetitions: nPr = n^r and nCr = (n+r−1)! / (r! × (n−1)!). The calculator applies the correct formula based on your repetition setting.
A combination is a selection of r items from a set of n items where the order of selection does not matter. For example, choosing 2 colours from {Red, Green, Blue} gives 3 combinations: {R,G}, {R,B}, and {G,B}.
All n distinct objects can be arranged in n! (n factorial) ways. For example, 4 objects can be arranged in 4! = 24 different ordered sequences.
Without repetitions means each item can only be selected once per arrangement or combination — like drawing cards from a deck without replacing them. With repetitions means the same item can appear more than once, like rolling a die multiple times.
Enter n (the total number of items in your set) and r (how many you want to choose). Select whether repetitions are allowed, and the calculator instantly displays both nPr and nCr results.
Because the order of numbers entered matters — entering 1-2-9 is different from 2-9-1. By definition, when order matters, we are dealing with permutations, not combinations. The common term 'combination lock' is a popular misnomer.
Without repetitions, r cannot exceed n — you cannot choose more items than exist in the set. If r > n with no repetitions, the result is 0. With repetitions allowed, r can be any positive integer regardless of n. You might also find our Venn Diagram Generator (3 Sets) useful.