Enter n (total number of items) and r (items to select) to calculate C(n,r) — the number of unique combinations possible when order doesn't matter. You also get the corresponding permutation count (nPr) and key factorials as supporting results.
Combinations C(n,r)
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Permutations P(n,r)
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n! (n Factorial)
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r! (r Factorial)
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(n−r)! Factorial
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A combination is a selection of r items from a set of n distinct items where the order of selection does not matter. For example, choosing 2 fruits from {apple, banana, cherry} gives the same combination regardless of which fruit you pick first. This differs from permutations, where the order does matter.
The formula is C(n, r) = n! / (r! × (n − r)!), where n is the total number of items, r is the number chosen, and '!' denotes factorial. For example, C(5, 2) = 5! / (2! × 3!) = 10, meaning there are 10 ways to choose 2 items from 5.
In a combination, the order of selected items does not matter (e.g., {A, B} = {B, A}). In a permutation, order does matter (AB ≠ BA). Permutations always produce a larger or equal count than combinations for the same n and r.
Multiply the combination result by r! (r factorial). So P(n, r) = C(n, r) × r!. For example, if C(5, 2) = 10 and r! = 2! = 2, then P(5, 2) = 20.
Divide the permutation count by r!. So C(n, r) = P(n, r) / r!. This accounts for all the different orderings of the same subset that permutations count separately but combinations treat as identical.
No. You cannot choose more items than exist in the set. If r > n, the combination is mathematically undefined (or considered 0). This calculator will flag that case and return 0.
The handshake problem asks: if n people are in a room and each person shakes hands with every other person exactly once, how many handshakes occur? The answer is C(n, 2) = n(n − 1) / 2. For 10 people, that's C(10, 2) = 45 handshakes.
JavaScript's floating-point numbers cannot accurately represent factorials beyond 170! (which already exceeds 10^306, near the limit of a 64-bit float). Inputs above 170 would produce Infinity or incorrect results, so the calculator caps n and r at 170 for reliable output.