Calculate permutations P(n,r) and combinations C(n,r) — with and without repetition. Find how many ways to arrange or select r items from n distinct items.
| Type | Formula | Order matters? | Repetition? |
|---|---|---|---|
| Permutation | n! / (n−r)! | Yes | No |
| Combination | n! / (r!(n−r)!) | No | No |
| Perm (rep) | nʳ | Yes | Yes |
| Comb (rep) | C(n+r−1, r) | No | Yes |
| Property | Calculation | Value |
|---|---|---|
| P(10,3) | 10! / (10−3)! = 10! / 7! | 720 |
| C(10,3) | 10! / (3! × 7!) | 120 |
| Pᵣₑₚ(10,3) | 10^3 | 1,000 |
| Cᵣₑₚ(10,3) | C(12,3) | 220 |
| n\r | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | |||||||
| 1 | 1 | 1 | ||||||
| 2 | 1 | 2 | 1 | |||||
| 3 | 1 | 3 | 3 | 1 | ||||
| 4 | 1 | 4 | 6 | 4 | 1 | |||
| 5 | 1 | 5 | 10 | 10 | 5 | 1 | ||
| 6 | 1 | 6 | 15 | 20 | 15 | 6 | 1 | |
| 7 | 1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 |
Calculate permutations (order matters) and combinations (order doesn't matter) instantly with our free calculator, useful for probability, statistics, and combinatorics problems.
Permutations: nPr = n! / (n-r)!. Combinations: nCr = n! / (r! × (n-r)!). The difference is whether arrangement order matters (permutations) or only the selection matters (combinations).
Permutations count arrangements where order matters (like race finishing positions); combinations count selections where order doesn't matter (like choosing a committee).
Because combinations group together all the different orderings of the same selection into one count, while permutations count each ordering separately, combinations essentially divide out the redundant orderings.
By mathematical convention, 0! = 1, this makes the permutation and combination formulas work correctly at the boundary cases.
Common applications include probability calculations, password/combination lock possibilities, lottery odds, and team or committee selection problems.