Permutations Calculator P(n, r)

P(n, r) = n! / (n−r)!. Number of ways to arrange r items from n where order matters.

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How to use this calculator

  • Enter n and r.
  • Read permutations.

About this calculator

Permutations count ordered arrangements. P(n,r) = n! / (n-r)! = n × (n-1) × … × (n-r+1). For r = n: P(n,n) = n! (all orderings of n items). Used when sequence matters: race orderings, password permutations, seating arrangements. Always P(n,r) ≥ C(n,r) by factor of r!.

Frequently asked

When does order matter?+
Race finish (1st, 2nd, 3rd are different roles). Passwords (123 ≠ 321). Seating (Alice next to Bob vs. far apart).
When doesn't it?+
Lottery (just need to match the numbers). Committees (5 members same regardless of pick order). Card hands (same hand, any deal order).
P(n,n) = n!?+
Yes — full permutations. 5 books on a shelf: 5! = 120 orderings.
Birthday paradox?+
Uses both. Probability all n people have different birthdays = P(365, n) / 365^n. Becomes <50% at n=23.
How to compute big P?+
Use product, not factorial division: n × (n−1) × … (r terms). Avoids overflow.

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