List 7. Let $x+y+z+w=20$. (8) Try to find (i) the number of positive integer solutions to equation (8); the number of groups;
(iii) the number of positive integer solutions to (8) satisfying $x \geq 7, y=8$;
(iv) the number of positive integer solutions to (8) satisfying $x \leqslant 6$;
(v) the number of positive integer solutions to (8) satisfying $x \leqslant 6, y \leqslant 7$.
(i) From the last statement of Example 4, the number of positive integer solutions of equation (8) is $C_{2}^{4-1}=C_{1}^{3}=969$.
(ii) Let $x^{\prime}=x-6$, then equation (8) becomes $x^{\prime}+y+z+w=14$.

Now, we find the number of positive integer solutions of (9) under the condition that $x \geqslant 7$ for equation (8). Therefore, the number of positive integer solutions is $C_{1}^{4}-1-1=C_{13}^{3}=286$.
(iii) Let $x^{\prime}=x-6, y^{\prime}=y-7$, then equation (8) becomes
$$
x^{\prime}+y^{\prime}+z+w=7.
$$

By the reasoning in (ii), the number of positive integer solutions is $C_{4-1}^{4}=C_{6}^{3}=20$.
(iv) Let $S_{1}=\{(x, y, z, w) \mid x \geq 7$, and $(x, y, z, w)$ is a positive integer solution of (8)\}, then the number of elements in $S_{1}$ (denoted as $\left|S_{1}\right|$) is $\left|S_{1}\right|=286$. From this, we deduce that the number of positive integer solutions of equation (8) satisfying $x \geq 6$ is $969-286=683$.
$$
\text{(v) Let } S_{2}=\{(x, y, z, w) \mid y \geqslant 8 \text{, }
$$
and $(x, y, z, w)$ is a positive integer solution of (8)\}, $S_{3}=\{(x, y, z, w) \mid x \geqslant 7, y \geqslant 8$, and $(x, y, z, w)$ is a positive integer solution of (8)\}, then $\left|S_{2}\right|=C_{13-1}^{1-1}=C_{12}^{3}=220, \quad\left|S_{3}\right|=20$. Let $I$ be the set of positive integer solutions of (8), then $|I|=969$. Now, from Figure 2, the number of positive integer solutions of (8) satisfying $x \geq 7, y \geq 8$ is
$$
|I|-\left|S_{1} \cup S_{2}\right|.
$$

But $\mid S_{1}\left\lfloor\left|S_{2}\right|=\left|S_{1}\right|+\left|S_{2}\right|-\left|S_{1} \cap S_{2}\right|\right|$. Also, note that $\left|S_{3}\right|=\left|S_{1} \cap S_{2}\right|$, so the number of solutions is
$$
\begin{aligned}
& |I|-\left|S_{1}\right|-\left|S_{2}\right|+\left|S_{3}\right| \\
= & 969-286-220+20=483 .
\end{aligned}
$$

As an exercise, the reader can find the number of positive integer solutions of (8) satisfying $x \leq 6, y \leqslant 7, z \leqslant 8, w \leqslant 9$ (Answer: 217).