Permutation Calculator

A permutation P(n,r) counts the ways to arrange r items chosen from n items where ORDER MATTERS. Picking a president, vice-president, and treasurer from 10 people is P(10,3) = 720. Here president Alice / VP Bob / treasurer Carol is a DIFFERENT outcome from Carol / Bob / Alice, so both are counted.

P(n,r) = n! / (n-r)!. For P(10,3): 10! / 7! = (10 x 9 x 8) = 720. You multiply the top r descending factors of n. When r = n it becomes a full arrangement: P(n,n) = n!.

Order. Permutations count arrangements where order matters (rankings, race finishes, passwords, assigned roles). Combinations count selections where order does not matter (committees, lottery numbers, poker hands). P(n,r) is always greater than or equal to C(n,r), and P(n,r) = C(n,r) x r!.

On a scientific calculator: enter n, press the nPr button, enter r, press =. On graphing calculators look under MATH > PRB > nPr. In Excel use PERMUT(n,r). In Python use math.perm(n,r).

Arranging books: how many ways to line up 5 of 8 books on a shelf? P(8,5) = 8! / 3! = 6,720. Race medals: how many ways can gold, silver, and bronze be awarded among 12 runners? P(12,3) = 1,320. Both matter because order (the exact position) changes the outcome.