How to use this calculator
- Enter n, the total number of items to choose from.
- Enter r, how many you choose.
- Use combinations when order doesn't matter (a lottery draw) and permutations when it does (a race podium).
Formula
Permutations count ordered arrangements; combinations count unordered selections.
nPr = n! / (n − r)!
nCr = n! / (r! (n − r)!)
With repetition: nʳ (ordered), (n + r − 1)Cr (unordered)
Examples
- Choosing 3 of 10 people for a committee: 10C3 = 120.
- Gold, silver, bronze among 10 runners: 10P3 = 720.
- 6 numbers from 49 in a lottery: 49C6 = 13,983,816.
Frequently asked questions
How do I know if order matters?
Ask whether swapping two chosen items gives a different outcome. If yes (ranking, PIN codes, seating), use permutations. If no (teams, hands of cards), use combinations.
What does 'with repetition' mean?
Items can be chosen more than once, such as digits in a PIN or scoops of the same ice-cream flavour.
Why is nCr the same as nC(n − r)?
Choosing r items to include is the same as choosing n − r items to leave out.