Stars and bars

Statement

Let $n$ and $k$ be natural numbers with $n \ge 1$. Then the number of multisets of size $k$ with elements in an $n$-element set is $$\left(\!\binom{n}{k}\!\right) = \binom{n + k - 1}{k}.$$ Equivalently, this is the number of solutions $(x_1, \dots, x_n)$ of $x_1 + \dots + x_n = k$ in natural numbers.

Remarks

The proof writes a multiset as a row of $k$ stars and $n - 1$ bars. The bars separate the stars into $n$ groups, one group for each element. The number of solutions in positive integers is $\binom{k - 1}{n - 1}$ for $k \ge 1$. This is the number of compositions of $k$ into $n$ parts.

Explanations

L1 human

To share $k$ identical sweets among $n$ children, put the $k$ sweets in a row and insert $n - 1$ dividers. There are $k + n - 1$ positions, and you choose which $k$ of them are sweets.