Calculate C(n,r) and combinations with repetition. Formula, steps, and BigInt result.
A combination counts the ways to choose r elements from a set of n when order does NOT matter, {A, B} and {B, A} are the same choice. Without repetition the count is C(n, r) = n! / (r! · (n − r)!), read 'n choose r' and written as a binomial coefficient. With repetition (a multiset, where the same element may be picked more than once) the count is CR(n, r) = C(n + r − 1, r). To stay exact, the tool evaluates the binomial with a multiplicative loop in BigInt instead of building giant factorials, and it uses the symmetry C(n, r) = C(n, n − r) to keep the loop short. Combinations underpin lottery odds, poker-hand counting, committee selection, and the coefficients of the binomial expansion.
Without repetition, the binomial coefficient is C(n, r) = n! / (r! · (n − r)!). Because order does not matter, we divide the permutation P(n, r) by the r! redundant orderings of each selection, that is, C(n, r) = P(n, r) / r!. The tool evaluates the binomial multiplicatively and exploits C(n, r) = C(n, n − r), avoiding huge intermediate numbers.
With repetition, we count multisets of size r from n available types: CR(n, r) = C(n + r − 1, r). Everything is computed with arbitrary-precision integers (BigInt), exact up to the 10¹⁰⁰ display limit.
Example 1, without repetition: in how many ways can you choose 2 people from a group of 5 (order irrelevant)? C(5, 2) = (5 · 4) / (2 · 1) = 10.
Example 2, real-world: the Mega-Sena lottery picks 6 numbers from 60, so there are C(60, 6) = 50,063,860 possible tickets, that is the chance of a single ticket winning.
Always C(n, 0) = 1 (one way to choose nothing) and C(n, n) = 1 (one way to choose everything). Without repetition, r cannot exceed n; with repetition, r may be larger than n, and the result climbs quickly toward the 10¹⁰⁰ overflow.
Combinations appear in probability and lotteries, counting card hands, biology (gene combinations), forming committees, and in the coefficients of Pascal's triangle and the expansion of (a + b)ⁿ.
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<iframe
src="https://jkit.tools/embed/en-US/combination-calculator"
width="100%"
height="600"
style="border:0"
loading="lazy"
title="Combination Calculator"
></iframe>These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
A combination is a selection of r elements from a set of n where the order of the chosen elements does not matter. For example, choosing 2 fruits from {apple, pear, grape} gives 3 combinations: {apple, pear}, {apple, grape}, and {pear, grape}. The count is given by C(n, r) = n! / (r! · (n − r)!).
C(n, r) = n! / (r! × (n−r)!)
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