Permutation & Combination Calculator

Free online permutation and combination calculator. Compute nPr (permutations), nCr (combinations), and factorials instantly.

Permutations count ordered arrangements (the order matters), while combinations count unordered selections (the order doesn't matter). If you're picking r items from n total, nPr = n!/(n−r)! counts arrangements and nCr = n!/(r!×(n−r)!) counts groups.

When to Use Each

  • Permutation (nPr): Passwords, race standings, seating arrangements — order matters
  • Combination (nCr): Lottery numbers, committee selection, card hands — order doesn't matter
  • Example: Picking 3 colors from red, green, blue, yellow — C(4,3)=4 combinations, P(4,3)=24 arrangements

Pascal's Triangle & Binomial Theorem

The combination C(n,r) corresponds exactly to the (r+1)th entry in the (n+1)th row of Pascal's Triangle. The Binomial Theorem states (a+b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ. Combinations appear throughout probability, statistics, and algebra.

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