Permutation & Combination Calculator

A permutation (nPr) is the number of ways to choose and arrange r items from n distinct items, where order matters. A combination (nCr) is the number of ways to choose r items where order does not matter. These are fundamental concepts in probability and statistics.

Enter the total number of items (n) and the number to choose (r) to calculate both nPr and nCr at once.

For example, the number of ways to choose 2 of 5 people in order (5P2) is 20, and the number of ways without order (5C2) is 10.

Permutation & Combination Calculator Calculation Method

Permutations (nPr): 0

Combinations (nCr): 0

About This Calculator

Permutations and combinations count the ways to choose items from a group. A permutation cares about order, a combination does not — which is why the combination count is always the smaller of the two.

Enter the total number of items (n) and how many you choose (r) to get nPr and nCr.

Permutations count ordered arrangements and combinations count unordered selections, and the ratio between them is exactly the number of orderings of the chosen items. Choosing 3 of 10 gives 120 combinations but 720 permutations, because each selection can be arranged in 3! = 6 ways. This distinction settles most counting problems.

Key Formula:nPr = n! / (n − r)! · nCr = n! / (r!(n − r)!)

Frequently Asked Questions

What is the difference between a permutation and a combination?

Permutations count ordered arrangements — 1-2-3 and 3-2-1 are different. Combinations count unordered selections — they are the same. A padlock code is a permutation; a lottery draw is a combination.

What are the odds of winning the lottery?

Choosing 6 numbers from 49 gives nCr = 13,983,816 possible tickets, so a single ticket has about a 1-in-14 million chance — roughly the chance of being struck by lightning in a given year.

How do I decide whether order matters?

Ask whether swapping two chosen items produces a different outcome. Medal winners are permutations, a committee is a combination, and a lottery ticket is a combination because the drawn numbers are unordered.

What changes when repetition is allowed?

Standard nCr excludes repetition. Allowing it gives (n + r − 1)Cr — the number of multisets of size r from n types, such as the ways to choose 3 scoops from 10 flavours.

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Sources & last update

2026-10-19