Counting arrangements and selections
A permutation is an arrangement where order matters: the ways to award gold, silver and bronze among 10 runners. A combination is a selection where order does not matter: the ways to pick a team of 3 from 10 players.
This calculator gives nPr and nCr for any n and r using exact big-number arithmetic, plus the counts when repeats are allowed.
How it is calculated
nPr = n! ÷ (n − r)!
nCr = n! ÷ (r! × (n − r)!)
With repetition: permutations n^r, combinations C(n + r − 1, r)
Worked example
Choosing 3 from 10
- Permutations 10P3
- 720
- Combinations 10C3
- 120
- 6 numbers from 49 (49C6)
- 13,983,816
Every combination of 3 items can be arranged in 3! = 6 orders, which is why 720 ÷ 6 = 120.
Permutation or combination?
- Ask: does the order change the outcome? If yes, use a permutation.
- Passwords, race positions and seating plans are permutations.
- Committees, lottery tickets and card hands are combinations.