Example 1.16 Find the number of different ways to distribute $r$ identical items to $n(n \leqslant r)$ people, such that each person gets at least one item.
Solution: Let the required number be $N$, and let the $i$-th person $(1 \leqslant i \leqslant n)$ receive $x_{i}$ items, then $x_{i} \geqslant 1$ and $x_{1}+x_{2}+\cdots+x_{n}=r$, so $N$ is equal to the number of positive integer solutions of the indeterminate equation $x_{1}+x_{2}+\cdots+$ $x_{n}=r$, which is $\binom{r-1}{r-n}$.