Combinations Calculator (nCr)

Combinations (nCr) count selections where order does not matter, like picking 3 toppings from a menu. Permutations (nPr) count arrangements where order matters, like awarding gold, silver, and bronze to 3 finalists. For any given n and r, nPr is always greater than or equal to nCr.

The standard formula is nCr = n! / (r! x (n - r)!). This calculator uses an iterative form that computes the product of (n - k + i) / i for i from 1 to k, where k = min(r, n - r), so division stays exact at every step and no enormous intermediate factorials are needed.

C(49, 6) = 13,983,816, so a single ticket has roughly a 1-in-14-million chance of matching all six numbers. Enter n = 49 and r = 6 to verify.

Lottery odds, poker hand counts, committee selection, clinical trial group assignments, and any problem where you choose a subset without caring about order. If the arrangement of the chosen items matters, use the permutations result the calculator also returns.

Yes. For n greater than 18, the calculator switches to BigInt arithmetic, so values like C(100, 50) = 100,891,344,545,564,193,334,812,497,256 compute exactly without any floating-point rounding.