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Permutation Calculator

A permutation is an ordered arrangement. Enter the total number of items n and how many you are choosing r to get nPr — the number of ways to pick and arrange them — along with the count when repeats are allowed and the matching number of combinations.

Quick examples

r cannot be larger than n when repeats are not allowed.

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Permutations (nPr)720
Formula
10! ÷ (10 − 3)! = 10! ÷ 7!
With repetition allowed (nʳ)
1,000
Combinations (nCr), if order did not matter
120

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      Formula

      nPr = n! ÷ (n − r)! = n × (n − 1) × … × (n − r + 1)
      With repetition allowed: nʳ
      nPr = nCr × r!

      How to use it

      1. Enter the total number of distinct items, n.
      2. Enter how many are chosen and arranged, r.
      3. Read nPr. Results over 30 digits are shown in scientific notation.

      Worked examples

      Arranging 3 of 10 items

      Permutations (nPr)
      720
      Formula
      10! ÷ (10 − 3)! = 10! ÷ 7!
      With repetition allowed (nʳ)
      1,000
      Combinations (nCr), if order did not matter
      120

      Gold, silver and bronze among 8 runners

      Permutations (nPr)
      336

      Dealing 5 cards in order from a 52-card deck

      Permutations (nPr)
      311,875,200
      Combinations (nCr), if order did not matter
      2,598,960

      Permutation or combination?

      Ask whether order matters. Awarding gold, silver and bronze is a permutation, because the same three runners in a different order is a different result. Choosing three people for a committee is a combination. There are always r! times as many permutations as combinations.

      Examples

      10P3 = 10 × 9 × 8 = 720. A 4-digit PIN with no repeated digit has 10P4 = 5,040 possibilities; if digits can repeat it has 10⁴ = 10,000. Arranging all n items is simply n!: five books fit on a shelf in 5! = 120 orders.

      Questions people ask

      What is 10P3?

      720, from 10 × 9 × 8.

      How many ways can 8 runners finish first, second and third?

      336, from 8 × 7 × 6.

      How many ways can 5 books be arranged on a shelf?

      120, which is 5! = 5 × 4 × 3 × 2 × 1.

      Related calculators

      Combination CalculatornCr — ways to choose r items from n when order does not matter.Factorial Calculatorn! exactly for small n, in scientific notation up to 5,000!.Probability CalculatorCombine two independent events: both, either, neither, exactly one.Binomial Distribution CalculatorExact and cumulative binomial probabilities for n trials.