Permutation Calculator

This permutation calculator finds nPr — the number of ways to arrange r items chosen from n when order matters — exactly, with the formula and every step shown.

Values
Permutations (nPr) —

How to Calculate Permutations

A permutation counts ordered arrangements — where the order of the chosen items matters.

1

Enter n and r

Type the total number of items n and how many you are arranging r. Both must be non-negative whole numbers, with r no larger than n.

2

The calculator applies the formula

It computes nPr = n! / (n − r)!, which multiplies out to n × (n−1) × … down to (n−r+1) — r factors in all. The result is exact, even for large numbers, using BigInt arithmetic.

3

Read the result and steps

The number of ordered arrangements appears first, with the expanded product below. If order does not matter, use the Combination Calculator instead.

Frequently Asked Questions

What is a permutation?
A permutation is an arrangement of items in which the order matters. nPr counts how many different ordered arrangements you can make by choosing r items from a set of n. For example, the ways to award gold, silver and bronze among 8 runners is a permutation, because who comes first, second and third are different outcomes.
What is the difference between a permutation and a combination?
A permutation counts arrangements where order matters, while a combination counts selections where order does not. Choosing a president and a vice-president is a permutation; choosing a two-person committee is a combination. For the same n and r, nPr is always at least as large as nCr, because each combination corresponds to r! ordered permutations.
What is the permutation formula?
The number of permutations of r items chosen from n is nPr = n! / (n − r)!. In expanded form that is n × (n−1) × … × (n−r+1), a product of r terms counting down from n. When r equals n, nPr is simply n!.
Does this calculator handle large numbers exactly?
Exact integer arithmetic is used for every permutation count. The calculator works with BigInt values rather than floating-point numbers, so nPr is computed with no rounding error even when n and r are large and the result has hundreds of digits. Order-matters counting grows very fast, and there are 311,875,200 ways to deal an ordered hand of 5 cards from 52. Enter whole numbers with n greater than or equal to r to see the exact count and the steps.

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