How many paths of length $N$ through a complete graph pass a given number of times $k_n$ through each node $n$ ($\sum_n k_n = N$)?
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We have a graph with vertices $1,2,3\dots j$ ,we want to know the number of paths of length $N$ such that for $1\leq i \leq j$ we have that vertex $i$ appears $k_i$ times. We are told $N$ is in fact equal to the sum of the $k_i$'s.
Every such path can be seen as a word of length $N$ which has $k_i$ appearances of symbol $i$, for $1\leq i \leq j$.
