8. We consider inequalities $n_{1}+\ldots+n_{N} \leqslant n$ with integer non-negative or positive $n_{i}, i=1, \ldots, N$. Count the number of ordered vectors $\left(n_{1}, \ldots, n_{N}\right)$ that satisfy the inequalities $n_{1}+\ldots+n_{N} \leqslant n, n \geqslant 1, N \geqslant 1$.
Solution. The number of ordered non-negative solutions to the inequality $n_{1}+\ldots+n_{N} \leqslant n$ is the same as the number of non-negative solutions to the equation $n_{1}+\ldots+n_{N}+n_{N+1}=n$, since specifying the first $N$ values uniquely determines the value of $n_{N+1}$. Therefore, we have $C_{N+n}^{n}$ solutions. In the case of finding positive solutions, we proceed as before - by subtracting one from each term in a positive solution, we obtain a non-negative solution to a similar inequality with $n^{\prime}=n-N$. Thus, we get $C_{n}^{n-N}$ positive solutions to the given inequality.