Permutations Calculator (nPr)
Find how many ordered arrangements of r items can be made from a set of n distinct items.
Ordered arrangements of r items chosen from n
- 1
Largest factor (n)
10 - 2
Smallest factor (n − r + 1)
10 − 3 + 1 = 8 - 3
P(n, r) = n × (n−1) × … × (n−r+1)
10 × 9 × 8 = 720
How does this calculator work?
Permutations count ordered arrangements of r items chosen from n distinct items without repetition, using P(n, r) = n! / (n − r)!. Enter n and r (whole numbers, 0 ≤ r ≤ n) to get the result. For example, P(10, 3) = 720. Order matters, so each distinct ordering is counted separately.
Formula
How this is calculated
A permutation counts the number of ways to arrange r items selected from a total of n distinct items when the order of selection matters and no item is repeated. Enter n (total items) and r (items chosen); both must be whole numbers satisfying 0 ≤ r ≤ n.
The formula P(n, r) = n! / (n − r)! simplifies to the falling product n · (n − 1) · ... · (n − r + 1), which has exactly r factors. This calculator evaluates that product multiplicatively rather than computing full factorials, which keeps intermediate values smaller and delays floating-point overflow for large n. For reference it also reports C(n, r) = P(n, r) / r!, the number of unordered selections (combinations).
Edge cases: P(n, 0) = 1 (one way to arrange nothing), P(n, n) = n! (a full ordering), and P(n, 1) = n. Results are integers; for very large inputs the value may exceed the exact-integer range of double-precision arithmetic and is shown in approximate form. Inputs with r > n, negatives, or non-integers are rejected.
Frequently asked questions
A permutation counts ordered arrangements, so ABC and ACB are different. A combination counts unordered selections, so ABC and ACB are the same. Permutations are always greater than or equal to the corresponding combinations: P(n, r) = C(n, r) × r!.
P(n, 0) equals 1. There is exactly one way to arrange zero items chosen from any set — the empty arrangement.
You cannot arrange more distinct items than you have. Since this formula assumes no repetition, r must satisfy 0 ≤ r ≤ n; otherwise the count is undefined (and one factor would be zero or negative).
Also known as
TG we-Calculate Editorial Team. (2026). Permutations Calculator (nPr) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/permutations-calculator
TG we-Calculate Editorial Team. "Permutations Calculator (nPr)." TG we-Calculate. 2026. https://we-calculate.com/calculator/permutations-calculator.
TG we-Calculate Editorial Team, "Permutations Calculator (nPr)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/permutations-calculator
@misc{wecalculate_permutations_calculator, title = {Permutations Calculator (nPr)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/permutations-calculator}}, year = {2026}, note = {TG we-Calculate} }
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