Combinations with repetition

Definition

Let $X$ be a set and let $k$ be a natural number. A $k$-combination with repetition of $X$ is a multiset of size $k$ with elements in $X$. That is a function $m : X \to \mathbb{N}$ with $\sum_{x \in X} m(x) = k$, where $m(x)$ is the multiplicity of $x$.

For natural numbers $n$ and $k$, the multiset coefficient $\left(\!\binom{n}{k}\!\right)$ is the number of $k$-combinations with repetition of an $n$-element set.