Permutation Calculator – P(n,r) Arrangements
Calculate the number of permutations P(n,r): ways to arrange r items from n distinct items when order matters. Enter n and r to get the result. Useful for probability, combinatorics, and counting problems.
Permutation Calculator
Calculate P(n,r) = n! / (n−r)! — the number of ways to arrange r items from n distinct items when order matters.
Permutation formula
P(n,r) = n! / (n−r)!
n = total items, r = items to arrange. Order matters. Example: P(5,3) = 5×4×3 = 60 ways to arrange 3 of 5 items in a row.
Understanding permutations and when to use them
A permutation counts the number of ways to arrange a subset of items when order matters. If you have 5 books and want to arrange 3 on a shelf, the order in which they appear matters: Book A, then B, then C is different from B, then A, then C. The number of such arrangements is P(5,3) = 5!/(5-3)! = 120/2 = 60.
The formula P(n,r) = n! / (n-r)! works because you have n choices for the first position, n-1 for the second, n-2 for the third, and so on until you have filled r positions. That product equals n! / (n-r)!. When r = n, you get P(n,n) = n!, which is the number of ways to arrange all n items.
Permutations appear in passwords (how many 4-digit PINs from 0-9? P(10,4) = 5040 if digits can repeat, but that is actually 10^4; P is for no repetition), race rankings (how many ways can 8 runners finish 1st, 2nd, 3rd? P(8,3) = 336), and many probability problems. Use our permutation calculator for quick results. For selections where order does not matter, use the combination calculator instead.