Example 2.3.6 Find the number of shortest paths from point $O(0,0)$ to point $P(10,5)$ in a coordinate grid that do not pass through any of the segments $AB, CD, EF, GH$, where the coordinates of $A, B, C, D, E, F, G, H$ are $A(2,2), B(3,2)$,
$$
\begin{array}{l}
C(4,2), D(5,2), E(6,2), F(6,3), \\
G(7,2), H(7,3) .
\end{array}
$$
Solution: Let $I$ be the set of all paths from $O$ to $P$;
$A_{1}$ be the set of all paths from $O$ to $P$ passing through $A B$;
$A_{2}$ be the set of all paths from $O$ to $P$ passing through $C D$;
$A_{3}$ be the set of all paths from $O$ to $P$ passing through $E F$;
$A_{4}$ be the set of all paths from $O$ to $P$ passing through $G H$.
Among them, $\left|A_{3} A_{4}\right|=0$, similarly $\left|A_{1} A_{3} A_{4}\right|=\left|A_{2} A_{3} A_{4}\right|=\left|A_{1} A_{2} A_{3} A_{4}\right|=0$,
$$
\begin{aligned}
\left|\overline{A_{1}} \overline{A_{2}} \overline{A_{3}} \overline{A_{4}}\right|= & |I|-\left|\bigcup_{i=1}^{4} A_{i}\right| \\
= & |I|-\left|A_{1}\right|-\left|A_{2}\right|-\left|A_{3}\right|-\left|A_{4}\right|+\left|A_{1} A_{2}\right| \\
& +\left|A_{1} A_{3}\right|+\left|A_{1} A_{4}\right|+\left|A_{2} A_{3}\right|+\left|A_{2} A_{4}\right| \\
& -\left|A_{1} A_{2} A_{3}\right|-\left|A_{1} A_{2} A_{4}\right| \\
= & \mathrm{C}_{15}^{5}-\mathrm{C}_{4}^{2} \mathrm{C}_{10}^{3}-\mathrm{C}_{6}^{2} \mathrm{C}_{8}^{3}-\mathrm{C}_{8}^{2} \mathrm{C}_{6}^{2}-\mathrm{C}_{9}^{2} \mathrm{C}_{5}^{2}+\mathrm{C}_{4}^{2} \mathrm{C}_{8}^{3} \\
& +\mathrm{C}_{4}^{2} \mathrm{C}_{6}^{2}+\mathrm{C}_{4}^{2} \mathrm{C}_{5}^{2}+\mathrm{C}_{6}^{2} \mathrm{C}_{6}^{2}+\mathrm{C}_{6}^{2} \mathrm{C}_{5}^{2}-\mathrm{C}_{4}^{2} \mathrm{C}_{6}^{2}-\mathrm{C}_{4}^{2} \mathrm{C}_{5}^{2} .
\end{aligned}
$$