Example 1.15 Let $N$ denote the number of different ways to distribute $r$ identical items to $n$ people, find $N$.


Solution: Let the $i$-th person $(1 \leqslant i \leqslant n)$ receive $x_{i}$ items, then $x_{i} \geqslant 0$ and $x_{1}+x_{2}+\cdots+x_{n}=r$, so $N$ is equal to the number of non-negative integer solutions of the indeterminate equation $x_{1}+x_{2}+\cdots+x_{n}=r$, which is $\binom{n+r-1}{r}$.