Skip to main content
Calculator Answers

Permutations Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A permutations calculator is a combinatorial tool that computes the total number of ordered arrangements of size r that can be formed from a set of n unique elements. Unlike combinations, the order of elements in permutations is significant. The calculator supports both arrangements without repetition and arrangements with repetition allowed. Permutations are heavily used in cryptography, passwords, coding theory, statistical probability, and orderings.

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).

Quick Answer

Calculate nPr permutations for ordered sets. Enter the total items and sample size to determine the number of distinct possible arrangements.

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 permutations calculator 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

Permutation CalculatorCalculate P(n, r) for arrangements where order matters. Supports without-repetition and with-repetition modes.

Frequently asked questions

Permutations are arrangements 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).