Puzzles/Statistical puzzles/Summing n

Puzzles | Statistical puzzles | Summing n

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Given a target sum $n$, you can choose $k$ summands each of which is a number, $n_{i}\in \{0,1,...\},i=1,...,k$, such that $\sum _{i=1}^{k}{n_{i}}=n$. How many ways are there of doing this?

Here the notion of a sum is same as that of a permutation, so two sums are same $iff$ they contain the same summands in the same order. E.g. $2+3+1$ and $1+3+2$ are not the same.

While you are at it, whats the answer, if $n_{i}\in \{1,2,...\}$?

solution