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$\text{clearly all the $a$'s must be such that $0 \leq a_k \leq n$}\\ \text{given any set of $k$ integers there is exactly one unlabelled ordering of them satisfying above}\\ \text{so we can choose $k$ integers $0\leq a_k \leq n$ and sort them, and label them accordingly}\\ \text{there are $N=(n+1)^k$ different sets of $k$ integers in the range $[0,n]$}$