Example 3-18 For the problem
$$
\begin{array}{c}
x_{1}+x_{2}+x_{3}=15 \\
0 \leqslant x_{1} \leqslant 5, \quad 0 \leqslant x_{2} \leqslant 6, \quad 0 \leqslant x_{3} \leqslant 7
\end{array}
$$

Find the number of integer solutions.