Permutations Calculator
Compute P(n,r) = n! / (n−r)! when order matters. Useful for seating, passwords, and race-place problems.
Permutations Calculator
Calculate P(n,r) - the number of ways to arrange r items from n items where order matters
What permutations are
A permutation is an ordered arrangement. ABC and BAC count as different. Combinations ignore order: {A,B,C} is one set either way.
Formula
P(n,r) = n! / (n−r)!
n items available, arrange r of them without reuse.
Example
3 places from 5 people:
P(5,3) = 5×4×3 = 60
Variants
- With repetition: n^r choices (each slot picks from all n)
- Identical items: n! / (n₁! n₂! …) so you do not overcount duplicates
- Circular: (n−1)! when rotations of a ring count as the same
Restrictions ("A and B must sit together") usually mean glue those people into one block, then permute the blocks. "Cannot sit together" is often total minus the glued case.
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Frequently Asked Questions
Permutations vs combinations?
Order matters for permutations; it does not for combinations. Race places → permutations. Lottery numbers → combinations.
Repetition?
Identical objects: divide by factorials of each repeated group. Free reuse each position: use n^r instead of P(n,r).
When use circular permutations?
When seating around a table and only relative position matters: (n−1)! for n distinct people.
Restrictions?
Glue forced neighbors into one unit, or subtract forbidden arrangements from the unrestricted total.
How large can n! get?
Factorials grow fast. Many calculators stop around 170! in floating point. Prefer canceling factors in P(n,r) before expanding full factorials.
Sanity-check an answer?
It should be ≤ n! (or n^r with repetition), ≥ 1 if any arrangement exists, and match a tiny hand-worked case of the same method.
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