Enter n (total objects) and r (objects chosen) to calculate nPr — the number of permutations. The Permutation Calculator applies the formula P(n,r) = n! / (n−r)! and returns the total number of ordered arrangements possible. Unlike combinations, order matters here — choosing A then B is counted separately from B then A. Also try the Odds Calculator.
Permutations P(n,r)
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n! (n Factorial)
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(n−r)! Factorial
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A permutation is an ordered arrangement of objects selected from a set. Unlike combinations, the order in which you select items matters — selecting A, B, C is different from selecting C, B, A. Permutations count every distinct ordering as a unique outcome. See also our Formula Classification — Truth Table Generator.
The formula is P(n, r) = n! / (n − r)!, where n is the total number of distinct objects and r is the number of objects being chosen. The exclamation mark (!) denotes factorial — the product of all positive integers up to that number.
In a permutation, order matters — A,B and B,A are considered two different results. In a combination, order does not matter — A,B and B,A are the same result. For any given n and r, the number of permutations is always greater than or equal to the number of combinations.
When n = r, the formula simplifies to n! because (n − r)! = 0! = 1. This means you are counting all possible ordered arrangements of all n objects, which equals n factorial. You might also find our calculate Probability of a Shared Birthday, Probability All Birthdays Are Unique & People Needed for 50% Chance — Birthday Paradox useful.
No. You cannot choose more objects than are available in the set. If r > n, the permutation is undefined (or zero), because you cannot arrange more items than exist without allowing replacements.
When r = 0, P(n, 0) = n! / n! = 1. There is exactly one way to choose and arrange zero items from any set — by choosing nothing. This is a mathematically valid edge case.
A factorial of a non-negative integer n (written n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1.
Permutations appear whenever order matters: arranging contestants in race finishing positions, creating PIN codes or passwords, scheduling tasks in sequence, or determining the number of ways to assign distinct roles to people selected from a group.