Permutation Calculator
Count the ways to arrange r items chosen from n when the order matters, as an exact whole number.
Permutations
How we calculated this
Worked example
Inputs: n = 10 runners, r = 3 medal places. Formula: P(n, r) = n! ÷ (n − r)! = n × (n − 1) × … × (n − r + 1). Substitution: 10 × 9 × 8. Result: P(10, 3) = 720 different ways to award gold, silver and bronze.
How it works
A permutation is an arrangement where the order matters: gold, silver and bronze to three different runners is a different result from the same three runners in a different order. P(n, r) counts the ordered arrangements of r items chosen from n. If the order does not matter, use the Combination Calculator, which counts selections instead. Each item can be used only once (there is no repetition), r cannot be more than n, and both must be whole numbers from 0 up, with n at most 1,000.
The result is always the exact whole number. It is worked out as a direct product of the r factors, not by writing out the full factorials, so it stays exact far beyond the size ordinary calculators can hold. Arranging nothing can be done in one way, so P(n, 0) = 1 and P(0, 0) = 1, and P(n, n) = n!. A very long result is summarised in the headline as a shorter "≈" figure; the summary is only a shorter way to write the same whole number, and the full exact digits are given under the result.
What this result means
The result is how many different ordered arrangements of r items can be made from n items. Two arrangements of the same items in a different order count separately. It is always an exact whole number.
Common mistakes
- Using permutations when the order does not matter. Dealing a five-card hand is a selection, so the Combination Calculator counts it, not this one.
- Allowing repeats. Each item can be used once only; arrangements that allow repeats are a different count.
Assumptions and limitations
- n and r must be whole numbers, r cannot be more than n, and n can be at most 1,000.
- A very long result is summarised in the headline; its full exact digits are always given below it.