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Permutation and Combination Calculator

How many ways to choose or arrange r items from n, with or without repeats, worked out exactly.

Rules last checked 25 October 2026Every step shown

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Combinations (order doesn't matter)

120

Permutations (order matters)
720

Choosing 3 from 10: 120 different groups, or 720 ordered arrangements.

How it's worked out

  1. Permutationsn! ÷ (n − r)! = 10! ÷ 7!720
  2. Combinationspermutations ÷ r! = 720 ÷ 3!120

The formulas

nPr = n! ÷ (n − r)! · nCr = n! ÷ (r! (n − r)!) · with repeats: nʳ and (n + r − 1)! ÷ (r! (n − 1)!)

Worked example

Choosing 3 of 10 people: 10 × 9 × 8 = 720 ordered line-ups, and 720 ÷ 3! = 120 different groups.

Questions people ask

What's the difference between a permutation and a combination?

In a permutation the order matters (first, second, third place); in a combination it doesn't (a team of three). Choosing 3 from 10 gives 720 permutations but only 120 combinations.

What is the formula for nCr?

nCr = n! ÷ (r! × (n − r)!). For 10 choose 3: 3,628,800 ÷ (6 × 5,040) = 120.

How many poker hands are there?

52 choose 5 = 2,598,960 different five-card hands.

Formula and rules last checked 25 October 2026Written and tested by the CalculatorDex teamReport a problem