Combinations & Permutations

Permutations count arrangements where order matters: choosing 1st, 2nd, 3rd place from 10 runners. Combinations count selections where order does not matter: choosing 3 team members from 10 people. Permutations of 10 choose 3 = 720; combinations of 10 choose 3 = 120. There are always more permutations than combinations.

nCr = n! / (r! x (n-r)!). For 10C3: 10! / (3! x 7!) = (10 x 9 x 8) / (3 x 2 x 1) = 720/6 = 120. The calculator shows the factorial expansion and simplification. nCr is also written as C(n,r) or "n choose r" and appears in the binomial theorem and Pascal's triangle.

nPr = n! / (n-r)!. For 10P3: 10! / 7! = 10 x 9 x 8 = 720. Permutations are always >= combinations because each combination generates r! different orderings. The calculator computes both values simultaneously so you can compare and choose the right formula for your problem.

Ask: "Does the order of selection matter?" If choosing lottery numbers (order does not matter): combinations. If assigning 1st, 2nd, 3rd place (order matters): permutations. Passwords (order matters): permutations. Committee selection (order does not matter): combinations. The calculator helps you identify which applies.

Combinations with repetition (where items can be selected more than once) use the formula: (n+r-1)! / (r! x (n-1)!). Example: choosing 3 scoops from 5 ice cream flavors (repeats allowed) = 7! / (3! x 4!) = 35. The calculator supports both with and without repetition for both combinations and permutations.