Combinations & Permutations Calculator
How many ways to choose or arrange r items from a set of n.
Combinations (nCr)
120
- Combinations (nCr)120
- Permutations (nPr)720
Inputs used
- Total items (n)10
- Items chosen (r)3
About this calculator
Combinations count selections where order doesn't matter; permutations count arrangements where it does. Choosing 3 pizza toppings from 10 is a combination (₁₀C₃ = 120); awarding gold, silver and bronze among 10 runners is a permutation (₁₀P₃ = 720).
The formulas: nPr = n! ÷ (n−r)!, and nCr = nPr ÷ r! — dividing by r! removes the orderings of the chosen items, which is why the permutation count is always r! times the combination count.
r can't exceed n (you can't choose more items than exist). The numbers explode quickly: a 52-card deck has more orderings (52!) than there are atoms visible in the universe.
Results are informational only — not financial, medical or legal advice.