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Permutation & Combination Calculator

Count how many ways you can choose or arrange items. Permutations care about order; combinations do not.

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Quantum Combinatorics

High-Dimensional Set Theory & Arrangement Solver

About the Permutation & Combination Calculator

Count how many ways you can choose or arrange items. Permutations care about order; combinations do not.

How to use the Permutation & Combination Calculator

  1. Enter the Total Population (n) and the Items to Select (r).
  2. Press Calculate to see both nPr and nCr.

Formula and method

Permutations: nPr = n! ÷ (n − r)!

Combinations: nCr = n! ÷ (r! × (n − r)!)

Worked example

Choosing 3 people from 10: as ordered positions (a president, secretary and treasurer) there are 10P3 = 720 ways. As an unordered committee there are 10C3 = 120 ways.

Order or no order. Choosing and ordering 3 of 5 people gives 5P3 = 60 arrangements. Choosing a group of 3 from 5 with no order gives 5C3 = 10 groups.

When to use the Permutation & Combination Calculator

  • Counting the ways to arrange or choose items.
  • Solving lottery, password and seating problems.
  • Checking answers to counting questions.

Common mistakes to avoid

  • Using permutations when the order does not matter, or combinations when it does.
  • Forgetting that repeated items change the count.

Tips

  • Ask yourself: does order matter? If yes, use permutations; if not, use combinations.
  • nCr is always less than or equal to nPr.

Frequently asked questions

How many lottery combinations are there?
Choosing 6 numbers from 49 gives 49C6 = 13,983,816 combinations.
What is the difference between a permutation and a combination?
In a permutation the order matters (a race result). In a combination it does not (a committee).
What does n! mean?
The factorial: n × (n − 1) × ... × 1. For example 5! = 120.