Permutation and combination calculator
The number of ways to arrange or choose r items from n: nPr and nCr.
About this calculator
This permutation and combination calculator counts the ways to arrange or choose r items from a set of n. Permutations count arrangements where order matters, and combinations count groups where it doesn't.
It also gives the counts with repetition allowed, as exact whole numbers.
How it works
nPr = n! ÷ (n − r)! counts arrangements where order matters. nCr = n! ÷ (r! × (n − r)!) counts selections where it does not.
Last reviewed:
Worked example
- Total items (n): 10
- Items chosen (r): 3
Combinations (nCr): 120 (when order does not matter)
Common questions
What is the difference between a permutation and a combination?
Order. Choosing a president and vice president from 10 people is a permutation, 90 ways. Choosing 2 people for a committee is a combination, 45 ways.
How many lottery combinations are there?
For a draw of 6 numbers from 49, nCr = 13,983,816. Enter 49 and 6 to see it.
What does with repetition mean?
Items can be chosen more than once, like digits in a PIN. A 4-digit PIN has 10⁴ = 10,000 permutations with repetition.
The story behind it
Where the idea came from: Indian mathematicians, including Bhāskara II in the 1100s, wrote down rules for counting permutations and combinations.
Counting arrangements interested scholars long before modern maths: Indian texts counted the ways to combine flavours or poetic metres. In Europe, Pascal's triangle of 1654 made combinations easy to read off. Today the same counts underpin probability, cryptography and computer science.
Why you can trust this calculator
- It runs in your browser. Nothing you type into it is sent anywhere.
- The method is explained on this page, so you can check it yourself.
- Free to use, with no sign-up.
- Results are estimates, so please double-check anything important.
- Spotted a mistake? Tell us and we'll look into it.
After 15 years as a mathematics educator, I built CalculStore to give students, teachers, professionals and anyone working with numbers simple, reliable tools.
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