Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 4 / 6E / i / Solution

Use a stars and bars bijection. For a tuple, place $N$ stars in $r+1$ consecutive blocks of lengths $n_0,\dots ,n_r$, separated by $r$ bars. The bar positions form an $r$-element subset of $\{1,\dots ,N+r\}$. Explicitly,They satisfy $1\le b_1<\cdots <b_r\le N+r$. Conversely, such a subset determinesThese numbers are nonnegative and sum to $N$, proving that the two constructions are inverse. HenceZero-length blocks are permitted, so the bijection also covers $N=0$.