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  "body": "\n## Statement\n\nLet $n$ and $k$ be natural numbers with $n \\ge 1$.\nThen the number of multisets of size $k$ with elements in an $n$-element set is\n$$\\left(\\!\\binom{n}{k}\\!\\right) = \\binom{n + k - 1}{k}.$$\nEquivalently, this is the number of solutions $(x_1, \\dots, x_n)$ of $x_1 + \\dots + x_n = k$ in natural numbers.\n\n## Remarks\n\nThe 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.\nThe number of solutions in positive integers is $\\binom{k - 1}{n - 1}$ for $k \\ge 1$. This is the [[number of compositions|057H72B2MB]] of $k$ into $n$ parts.\n\n## Explanations\n\n### L1 human\n\nTo share $k$ identical sweets among $n$ children, put the $k$ sweets in a row and insert $n - 1$ dividers.\nThere are $k + n - 1$ positions, and you choose which $k$ of them are sweets.\n",
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