Probability & Combinations / Permutations Calculator

How many ways to choose 3 from 10, when order doesn't matter

C(10,3), choosing 3 items from a set of 10 where order doesn't matter, equals 120 -- calculated as 10! / (3! x (10-3)!) = 10! / (3! x 7!).

Combinations (C) and permutations (P) answer genuinely different questions: combinations count groupings where order is irrelevant (picking 3 people for a committee), while permutations count orderings where sequence matters (picking 1st, 2nd, and 3rd place from 10 runners) -- switching the mode from C to P on the same n and r values would produce a different, larger result (720 for P(10,3)), since permutations count each different ordering of the same 3 items separately.

120 as the result for C(10,3) can be sanity-checked against the permutation count: P(10,3) = 720, and since each unique group of 3 can be arranged in 3! = 6 different orders, 720 / 6 = 120 -- confirming combinations are permutations divided by the number of ways to order each group.

Practical limits on n and r

n and r are floored. r cannot exceed n. Large n can overflow JS numbers. Non-integer inputs for n or r are floored down to the nearest whole number before calculating, since combinations and permutations are only defined for whole numbers of items -- entering 10.5 would be treated as 10.

r (the number chosen) can never exceed n (the total pool) for either combinations or permutations to make logical sense -- you can't choose more items than exist in the set, so this calculator enforces that constraint rather than returning a nonsensical result.

Factorials grow extremely fast -- large values of n (particularly beyond the 20s) can produce numbers that exceed JavaScript's safe integer precision, which is worth knowing if you're working with a large population size, since the exact result could lose precision beyond a certain point.

Related combinatorics and probability tools

For a related structured numeric operation, the Matrix Add/Multiply Calculator handles a different kind of mathematical computation frequently used alongside combinatorics in coursework and applications.

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