Skip to main content
Calculator Answers

Permutation Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A permutation calculator computes the number of unique ways to arrange a subset of items selected from a larger set where the order of selection is significant. It calculates both permutations without repetition, using the formula P(n, r) = n! / (n-r)!, and permutations with repetition, using n^r. Statisticians, computer scientists, and combinations planners use this calculation to solve combinatorial problems, analyze security passcode possibilities, and determine potential sequences in probability experiments.

Pick without-repetition or with-repetition mode, enter the total number of items n and the number chosen r, and the calculator returns the count of permutations P(n, r).

See also: Permutations Calculator, Combination Calculator, and Factorial Calculator.

Quick Answer

Calculate the number of unique arrangements for a set of items where order matters. Enter the total items and number to select.

How many distinct items you have to choose from. · e.g. 10

Cannot exceed n in without-repetition mode. · e.g. 3

Formula

P(n, r) = n! ÷ (n − r)!

Permutations count arrangements where order matters: ABC is different from CAB. Use combinations instead when order does not matter.

Permutations

P(10, 3) without repetition

720

Order matters · P(n, r) = n! ÷ (n − r)!

n10
r3
Step10 × 9 × … × 8 (3 factors)
Result720
Was this helpful?

Examples

P(10, 3) without repetition

720 (10 × 9 × 8)

P(5, 5) without repetition

120 (5!)

P(4, 3) with repetition (4^3)

64

How it works

A permutation counts ordered arrangements. The without-repetition formula uses descending factorial: multiply n by (n − 1), (n − 2), and so on for r terms total. With repetition, each of the r positions is independent, giving n^r.

Without repetition · P(n, r) = n! ÷ (n − r)!

With repetition · n^r

Conventions: P(n, 0) = 1 · 0! = 1 · P(n, n) = n!

Related counting and math calculators

Related Calculators

More tools from Education

Permutations CalculatorCalculate nPr permutations for ordered arrangements. Supports arrangements with or without repetition.Combination CalculatorCalculate C(n, r) for unordered selections, using the multiplicative form for stability across small and large inputs.Factorial CalculatorCalculate n! for any non-negative integer up to 170 with full step-by-step expansion, supporting big numbers via BigInt.

Frequently asked questions

A permutation is an arrangement of items where order matters. ABC, ACB, BAC, BCA, CAB, and CBA are six different permutations of the same three letters. The standard P(n, r) counts how many ways you can arrange r items chosen from a set of n distinct items.

Without repetition, P(n, r) = n! ÷ (n − r)!. With repetition, the count is simply n raised to the r power (n^r), because each of the r positions can be any of the n items independently. The calculator offers both modes via the toggle.

Without repetition is the right mode when an item can only appear once in the arrangement: who finishes 1st, 2nd, 3rd in a race, or which 4 letters spell a word using each letter only once. With repetition is the right mode when items can repeat: 4-digit PIN codes where digits can repeat, or license plates where any letter can appear in any slot.

Permutations count arrangements where order matters; combinations count selections where order does not. Choosing the gold, silver, and bronze winners is a permutation. Choosing 3 finalists out of a pool is a combination. P(n, r) is always at least as large as C(n, r), and equals r! × C(n, r).

P(n, 0) = 1 by convention: there is exactly one way to arrange zero items (the empty arrangement). Similarly, 0! = 1 by convention, which makes the formula P(n, n) = n! ÷ 0! = n! consistent.

In without-repetition mode, that is impossible: you cannot pick more distinct items than the set contains, and the calculator flags it. In with-repetition mode, r can freely exceed n because each position is filled independently. A 6-character password from a 4-letter alphabet has r > n and is perfectly valid (4^6 = 4,096 possibilities).