【Example 2】Find the number of integer solutions to the equation $x+y+z=1$, with the requirement that $x, y, z$ are all greater than 16.
Since it is required that $x>-6, y>-6, z>-6$, this is equivalent to requiring $x+5 \geqslant 0, y+5 \geqslant 0, z+5 \geqslant 0$. The original equation is $(x+5)+(y+5)+(z+5)=16$, so the original problem can be transformed into finding the number of non-negative integer solutions to the equation
$$
u+v+w=16
$$

To solve the latter problem, we imagine placing 16 indistinguishable balls into 3 different boxes, with the number of balls placed in the 3 boxes being regarded as the values of $u, v, w$ respectively. It is evident that the number of different ways to place the balls is the same as the number of non-negative integer solutions to the equation, so we know that there are $C_{16+3-1}^{16}=C_{18}^{2}=153$ such solutions.