Combination

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 of an $n$-element set is the binomial coefficient $\binom{n}{k}$.

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$\}$.