Last updated: 2026-05-18

Definition

A combination $C(n, r)$ counts the number of ways to choose $r$ items from a set of $n$ distinct items when the order of selection does not matter. Choosing {A, B} is the same outcome as {B, A}.

The formula is:

This is also called the binomial coefficient and read as "$n$ choose $r$."

Worked example

How many 5-card hands can be dealt from a standard 52-card deck?

Nearly 2.6 million distinct hands from just 52 cards. The calculator above handles $n \leq 20$, but the formula itself extends to any $n$.

Combination versus permutation

The key question is: does order matter? Use the Permutation Calculator — P(n, r) when sequence matters.

The relationship: $C(n, r) = P(n, r) / r!$. A combination is a permutation with the $r!$ orderings of the selected group divided out.

Symmetry property

Choosing 3 items from 10 is equivalent to choosing which 7 items to leave out. This symmetry is useful for reducing computation: if $r > n/2$, use $C(n, n-r)$ instead.

Pascal's Triangle

Combinations follow a simple recurrence:

This identity builds Pascal's triangle, where each entry is the sum of the two entries directly above it. Row $n$ of Pascal's triangle lists $C(n, 0), C(n, 1), \ldots, C(n, n)$.

Common applications

• Lottery analysis: counting possible ticket combinations
• Team selection: choosing players from a squad
• Genetics: counting genotype possibilities from allele pools
• Quality control: selecting a sample from a batch for inspection
• Card games: counting hand possibilities in poker, bridge, etc.

Practical notes

Special cases. $C(n, 0) = C(n, n) = 1$ for any $n$. There is exactly one way to choose nothing, and one way to choose everything.

$r > n$ is undefined. You cannot choose more items than exist; the calculator reports an error for this case.

With repetition. The formula above assumes without replacement. The multiset coefficient $\binom{n+r-1}{r}$ handles the with-repetition case, but that is a separate calculation.