---
tag: 0JZSRWRBZE
kind: definition
name: Combination
msc: [05A05]
defines: 098PHYBTV9
sources:
  - url: https://en.wikipedia.org/wiki/Combination
    locator: lead section
---

## Definition

Let $X$ be a set and let $k$ be a natural number.
A $k$-combination of $X$ is a subset $S \subseteq X$ with $|S| = k$.

The set of all $k$-combinations of $X$ is written $\binom{X}{k}$.

## Remarks

A combination does not record an order of its elements. Two lists with the same elements in a different order give the same combination.
For $X = \{a, b, c\}$, the 2-combinations are $\{a, b\}$, $\{a, c\}$ and $\{b, c\}$.
The [[number of k-element subsets|0BFMR2FQ8B]] of an $n$-element set is the [[binomial coefficient|092GEF38YD]] $\binom{n}{k}$.

## Explanations

### L1 human

A combination is a selection of $k$ objects where the order does not matter.
Choosing a team of 2 from Ann, Bob and Cy gives 3 teams, because the team $\{$Ann, Bob$\}$ is the same as $\{$Bob, Ann$\}$.