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Permutation & Combination Calculator
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
- Enter the Total Population (n) and the Items to Select (r).
- 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
Choosing 6 numbers from 49 gives 49C6 = 13,983,816 combinations.
In a permutation the order matters (a race result). In a combination it does not (a committee).
The factorial: n × (n − 1) × ... × 1. For example 5! = 120.