How to calculate combinations
- Enter the total number of items, n, and how many you choose, r.
- Read the number of combinations (order does not matter) and permutations (order matters), with and without repetition.
- See the odds of getting one specific combination, which is how lottery odds are calculated.
nCr versus nPr
Combinations count selections where order does not matter: C(n, r) = n! ÷ (r!(n − r)!). Choosing a 3-person committee from 10 people gives C(10, 3) = 120 possibilities. Permutations count arrangements where order matters: P(n, r) = n! ÷ (n − r)!, so awarding gold, silver and bronze among 10 runners gives 720 possibilities, 3! = 6 times as many. With repetition allowed (like a PIN code), there are nʳ permutations and C(n + r − 1, r) combinations.
A 6-from-49 lottery has C(49, 6) = 13,983,816 combinations, so a single ticket wins the jackpot with odds of 1 in 13,983,816. The calculator uses exact big-integer arithmetic, so even huge results like C(1000, 500), a number with 300 digits, are exact; very long results are shown in scientific notation. Everything is calculated in your browser.
FAQ
What does nCr mean?
"n choose r": the number of ways to choose r items from n when order does not matter.
When should I use permutations?
When order matters, such as rankings, passwords or seating arrangements.
What are my odds in a 6/49 lottery?
1 in 13,983,816 for the jackpot.