Permutation Calculator
Count ordered arrangements: how many ways to pick r items from n where order matters.
What the Permutation Calculator does
A permutation counts arrangements where order matters — first, second and third place in a race are different outcomes. Fewer choices remain at each step, which is why the product runs down from n.
Formula
P(n, r) = n! ÷ (n − r)!With repetition: nʳ
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Total items (n) | number | Yes | Accepts 0 or more. |
| Items chosen (r) | number | Yes | Accepts 0 or more. |
How to use it
- Enter Total items (n) and Items chosen (r).
- Select Calculate.
Worked example
How many ways can gold, silver and bronze be awarded among 8 runners?
- n
- 8
- r
- 3
P(8,3) = 8 × 7 × 6 = 336 podium orders.
Reading the result
- Permutations count arrangements where order matters, so first and second place swapped is a different outcome.
- Permutation counts grow extremely fast: arranging just 10 items gives over 3.6 million orderings.
Common mistakes
- Using permutations where order is irrelevant, which overcounts by a factor of r! — that case needs combinations.
- Allowing repetition without meaning to. Drawing with replacement gives nʳ, not the falling product used here.
Frequently asked questions
How is a permutation different from a combination?
Permutations count order, combinations do not. ABC and CBA are two permutations but one combination.