Permutation & Combination Calculator
Compute permutations (nPr) and combinations (nCr) for choosing r items from a set of n.
Ordered arrangements of r items chosen from n.
- 1
n!
10! = 3,628,800 - 2
(n − r)!
(10 − 3)! = 7! = 5,040 - 3
Permutations nPr = n! ÷ (n − r)!
3,628,800 ÷ 5,040 = 720Ordered selections of r items from n. - 4
Combinations nCr = nPr ÷ r!
720 ÷ 6 = 120
Formula
How this is calculated
You enter two whole numbers: n is the total number of distinct items in the set, and r is how many of them you select. The calculator first requires 0 ≤ r ≤ n, since you cannot choose more items than exist. It then builds every result from the factorial n!, which is the product of all positive integers up to n (with 0! defined as 1).
Permutations count ordered selections, so nPr = n! / (n − r)! — the (n − r)! in the denominator cancels the arrangements you never use, leaving every distinct ordering of the r chosen items. Combinations count unordered selections, so nCr = n! / (r! × (n − r)!) takes that permutation count and divides by r!, the number of ways the same r items can be reordered. Because each combination corresponds to r! permutations, nCr is always ≤ nPr.
The inputs are pure counts with no units, and order is the only assumption that distinguishes the two formulas. The main limitation is numeric: factorials grow extremely fast, so for large n the exact integer results exceed standard floating-point precision and become approximate.
About this calculator
Permutations and combinations count the ways you can select items from a larger set. A permutation counts arrangements where order matters — picking 1st, 2nd and 3rd place from a race, for example. A combination counts selections where order does not matter — like choosing three lottery numbers or members of a committee.
Both are built from factorials. The permutation formula nPr = n! / (n − r)! counts ordered selections, while the combination formula nCr = n! / (r! × (n − r)!) divides out the r! ways each group can be ordered. Because nCr ignores order, it is always less than or equal to nPr. Both require 0 ≤ r ≤ n, and very large values of n can produce results too big to represent exactly.
Frequently asked questions
A permutation counts arrangements where the order of selection matters, while a combination counts groups where order does not matter. nCr is always smaller than or equal to nPr.
n! (n factorial) is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
You cannot choose more items than exist in the set. If r exceeds n the formulas are undefined, so the calculator requires 0 ≤ r ≤ n.
Also known as
TG we-Calculate Editorial Team. (2026). Permutation & Combination Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/permutation-combination-calculator
TG we-Calculate Editorial Team. "Permutation & Combination Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/permutation-combination-calculator.
TG we-Calculate Editorial Team, "Permutation & Combination Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/permutation-combination-calculator
@misc{wecalculate_permutation_combination_calculator, title = {Permutation & Combination Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/permutation-combination-calculator}}, year = {2026}, note = {TG we-Calculate} }
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