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for questions about multinomial coefficients
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Closed form for a binomial product sum
For $j = 0$, the summand ${n \choose i_1} \cdots {i_{k - 1} \choose i_k}$ is counting the number of length $k+1$ chains $\{1, \ldots, n\} =: S_0 \supsetneq S_1 \supsetneq S_2 \supsetneq \cdots \supset …