Permutation and Combination Calculator
Calculate the number of ways to arrange or choose items from a set.
Try an example
13,983,816
About this calculator
How This Permutation & Combination Calculator Works
Enter a total number of items (n) and how many are chosen (r), and the calculator returns both the number of combinations (order doesn't matter) and permutations (order matters).
Worked Example
Choosing 6 numbers from 49 (a classic lottery format): there are 13,983,816 possible combinations — and 10,068,347,520 permutations, since that count also treats every different ordering of the same 6 numbers as distinct.
The Formulas
Permutations: P(n,r) = n! ÷ (n−r)! — counts arrangements where order matters. Combinations: C(n,r) = n! ÷ [r!(n−r)!] — counts selections where order doesn't matter, which is why it's always smaller: it divides out all the different orderings of the same chosen group.
When to Use Which
- Use combinations when picking a group where order doesn't matter — lottery numbers, a committee, a hand of cards.
- Use permutations when order matters — a race's finishing order, a PIN code, seating arrangements.
Worked example
Choosing 6 numbers from 49 (like a lottery draw) gives C(49,6) = 13,983,816 possible combinations, where order doesn't matter — far more than the 10,068,347,520 permutations, where it does.
Frequently asked questions
What's the difference between a permutation and a combination?
A permutation counts arrangements where order matters (ABC is different from BCA); a combination counts selections where order doesn't matter (ABC and BCA are the same group).
Why is the combination count always smaller?
Every combination of r items can be arranged in r! different orders, all of which count as separate permutations — so permutations = combinations × r!.
What if r is larger than n?
That's not possible — you can't choose or arrange more items than exist in the set, so the calculator won't return a result.