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 $n_i \in \{1, 2, ...\}$?

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