Count how many ways r items can be chosen from n, as combinations when order does not matter and permutations when it does.
Calculations happen locally. We do not send your values to our server.
Choosing 2 people from 5 gives 10 combinations and 20 permutations. A 5-card hand from a 52-card deck has 2,598,960 combinations, and a lottery drawing 6 numbers from 49 has 13,983,816.
Combinations count selections where order is irrelevant, using n!/(r!(n−r)!) evaluated with integer cancellation so nothing is rounded. Permutations count ordered arrangements by multiplying n × (n−1) down through r positions. Every combination of r items can be arranged r! ways, so permutations always equal combinations × r!: 10 × 2 = 20.
Enter whole numbers with r no larger than n and n at most 1,000. The tool counts selections without repetition, so it does not cover cases where an item can be picked twice; a 4-digit PIN allows 10 × 10 × 10 × 10 = 10,000 possibilities instead.
Enter the number of available items n and how many are chosen r. Choosing 2 people from 5 for a committee gives 10 combinations, because Alex and Sam is the same pair as Sam and Alex. Choosing a chair and a secretary from the same 5 people gives 20 permutations, because the two roles make order matter. Every combination of r items can be arranged r! ways, so permutations are always combinations multiplied by r!: 10 × 2 = 20.
A 5-card poker hand from a 52-card deck has 2,598,960 possible combinations, while dealing those 5 cards in order gives 311,875,200 permutations. A lottery that draws 6 numbers from 49 has 13,983,816 combinations, so one ticket has a 1 in 13,983,816 chance of matching all six. Choosing 3 toppings from 6 gives 20 combinations and 120 ordered arrangements. The results are calculated with exact whole-number arithmetic, so large answers are not rounded.
Choosing zero items has exactly one combination and one permutation, the empty selection. Choosing all 10 of 10 items has one combination but 3,628,800 permutations, which is 10!. Inputs must be whole numbers with r no larger than n and n at most 1,000. This tool counts selections without repetition. If an item can be picked more than once, such as digits in a PIN, use repeated powers instead: a 4-digit PIN has 10 × 10 × 10 × 10 = 10,000 possibilities.
| Input | Result |
|---|---|
n 5 · r 2 | C 10 · P 20 |
n 6 · r 3 | C 20 · P 120 |
n 52 · r 5 | C 2,598,960 · P 311,875,200 |
n 49 · r 6 | C 13,983,816 · P 10,068,347,520 |
n 10 · r 0 | C 1 · P 1 |
n 10 · r 10 | C 1 · P 3,628,800 |
Use combinations when reordering the same items changes nothing, such as picking a committee of 2 from 5: Alex and Sam is the same pair as Sam and Alex, so there are 10. Use permutations when roles or positions matter, such as choosing a chair and a secretary, which gives 20.
Each extra item multiplies the count. Choosing 3 toppings from 6 gives 20 combinations, but dealing 5 cards in order from 52 gives 311,875,200 permutations. A lottery with 13,983,816 combinations means one ticket has a 1 in 13,983,816 chance of matching all six numbers.
Choosing zero items has exactly one combination and one permutation: the empty selection. Choosing all 10 of 10 items also has one combination, because there is only one possible set, but 3,628,800 permutations, which is 10 factorial, since every order counts separately.
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