Stars and bars

The number of nonnegative integer solutions of $x_1+\cdots +x_k=N$ is $\left(\genfrac{}{}{0ex}{}{N+k-1}{k-1}\right)$.

To count nonnegative tuples with ${\sum }_{i=0}^r{n}_i=N$ and $n_i<a_i$, mark violations $n_i\ge a_i$. An intersection of violations indexed by $J$ is in bijection with unrestricted tuples of sum $N-{\sum }_{i\in J}{a}_i$, by subtracting the bounds from those coordinates. Stars and bars counts this intersection; the inclusion-exclusion principle then gives the displayed formula. A negative remaining sum contributes zero.