The number of different multisets There are ${n \choose k}$ ways of choosing $k$ distinct items from a set of size $n$. There are $n^k$ ways of choosing $k$ items if we allow duplicates. How many distinct multi-sets a...--prophetes.ai

The number of different multisets There are ${n \choose k}$ ways of choosing $k$ distinct items from a set of size $n$. There are $n^k$ ways of choosing $k$ items if we allow duplicates. How many distinct multi-sets are there of size $k$? It must be considerably fewer than $n^k$ as, if the set $S = \\{1,2,3,4,5\\}$ then we say the multi-subsets $\\{1,1,2,3\\}$ and $\\{1,2,1,3\\}$, both of size $4$, are the same.

It is called multiset coefficient.

$$\left(\\!\\!{n\choose k}\\!\\!\right)=\binom{n+k-1}{k}$$

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