Permutations & Combinations Calculator

Count arrangements (permutations) and selections (combinations), with or without repetition, and see exact results even when they are enormous.

Your numbers

Items to choose from (n)

Items chosen (r)

Repetition allowed (items can be chosen more than once)

Combinations

120

nCr = n! ÷ (r! × (n − r)!) = 10! ÷ (3! × 7!)

(10 × 9 × 8) ÷ 3! = 720 ÷ 6 = 120

For reference: 10! = 3,628,800

How to use this tool

Choose permutations if the order of the chosen items matters (a race podium), or combinations if it does not (a committee).
Enter n, the number of items to choose from, and r, the number you choose.
Tick "Repetition allowed" if the same item can be picked more than once.
Read the exact result, the formula, and the working underneath.

Formula Reference

Permutations (no repetition)

nPr = n! ÷ (n − r)!

Combinations (no repetition)

nCr = n! ÷ (r! (n − r)!)

Permutations with repetition

nʳ

Combinations with repetition

C(n + r − 1, r)

ArticlePermutations and Combinations ArticleThe Birthday Problem ArticleThe Inclusion–Exclusion Principle CalcBasic Probability Calculator GlossaryGlossary: Factorial

Frequently Asked Questions

How do I know whether to use a permutation or a combination?+

Ask whether swapping two chosen items gives a different outcome. If it does, as with first, second, and third place, use permutations. If it does not, as with choosing three people for a committee, use combinations.

Why is nCr always smaller than nPr?+

Each combination of r items can be arranged in r! different orders, and a permutation counts every order separately. So nPr = nCr × r!.

What does "with repetition" mean?+

It means an item can be chosen again after it has been picked, such as the digits of a PIN code. Without repetition each item can be used at most once, as when dealing cards from a deck.

Can it handle very large numbers?+

Yes. The calculator uses exact whole-number arithmetic, so results such as C(52, 5) = 2,598,960 or far larger values are shown digit by digit, not rounded. Inputs are limited to 1,000.

How to use this tool

Choose permutations if the order of the chosen items matters (a race podium), or combinations if it does not (a committee).

Enter n, the number of items to choose from, and r, the number you choose.

Tick "Repetition allowed" if the same item can be picked more than once.

Read the exact result, the formula, and the working underneath.