Permutation & Combination Calculator

Calculate nPr (permutations) and nCr (combinations) for counting and probability problems.

Permutation & Combination

P(10,3) — Permutations

720

Order matters

C(10,3) — Combinations

120

Order doesn't matter

P(10,3) = 10! / (10-3)! = 720

C(10,3) = 10! / (3! × (10-3)!) = 120

Permutation & Combination Calculator Formula & How It Works

P(n,r) = n! / (n−r)! | C(n,r) = n! / [r!(n−r)!]
  • Permutation (nPr): ordered arrangement (order matters)
  • Combination (nCr): unordered selection (order doesn't matter)
  • n! = n × (n−1) × … × 2 × 1
  • C(n,r) = P(n,r) / r! (divide by r! to remove order)

Use permutations when order matters (race finishers, PIN codes, anagrams). Use combinations when order doesn't matter (lottery, team selection, card hands). C(n,r) is always ≤ P(n,r) because combination groups are the same as P(n,r)/r! ordered arrangements. C(52,5) = 2,598,960 possible poker hands.

Permutation & Combination Calculator FAQs

What is the difference between permutation and combination?

Permutation: order matters. Selecting 3 from 5 people for 1st/2nd/3rd place = P(5,3) = 60 ways. Combination: order doesn't matter. Selecting 3 from 5 people for a committee = C(5,3) = 10 ways. Same people, different count because committee {A,B,C} = {B,A,C}.

How do you calculate factorial?

n! = n × (n−1) × (n−2) × … × 2 × 1. 5! = 5×4×3×2×1 = 120. Special cases: 0! = 1 (by definition). 1! = 1. Factorials grow extremely fast: 20! ≈ 2.4 × 10¹⁸.

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