For example, $7 x, y, z$ are all positive integers, the equation $x+y+z=15$ has how many sets of solutions?
Let $x=x^{\prime}+1, y=y^{\prime}+1, z=z^{\prime}+1$, then the problem reduces to finding the non-negative integer solutions of $x^{\prime}+y^{\prime}+z^{\prime}=12$, whose solutions correspond to the generating function $\frac{1}{(1-x)^{3}}=\sum_{r=0}^{\infty} C_{3+r-1}^{r} x^{r}$. When $r=12$, $C_{3+12-1}^{12}=C_{14}^{2}=$ 91, which is the number of solutions we are looking for.