Example 4-17 Use 5 *s and 4 ○s to place on a $3 \times 3$ grid. How many different layouts are there, considering rotations and reflections as the same?
Substitution Group:
Format: (1), 1
$$
\begin{array}{l}
(1)^{3}(2)^{3}, 4 \text { } \\
(1)^{1}(4)^{2}, 2 \text { } \\
\text { (1) }(2)^{4}, 1 \text { }
\end{array}
$$
$$
\begin{aligned}
P(x)= & \frac{1}{8} \times\left[(1+x)^{9}+4(1+x)^{3}\left(1+x^{2}\right)^{3}\right. \\
& \left.+2(1+x)\left(1+x^{4}\right)^{2}+(1+x)\left(1+x^{2}\right)^{4}\right]
\end{aligned}
$$
(1) Corresponding to $(1)^{9},(1+x)^{9}$, the coefficient of the $x^{5}$ term is: $C(9,5)=126$;
(2) Corresponding to $(1)^{3}(2)^{3}, 4(1+x)^{3}\left(1+x^{2}\right)^{3}$, the coefficient of the $x^{5}$ term is:
$$
4[C(3,1) C(3,2)+C(3,3) C(3,1)]=48 ;
$$
(3) Corresponding to $(1)^{1}(4)^{2}, 2(1+x)\left(1+x^{4}\right)^{2}$, the coefficient of the $x^{5}$ term is:
$$
2 C(2,1)=4 \text {; }
$$
(4) Corresponding to $(1)(2)^{4},(1+x)\left(1+x^{2}\right)^{4}$, the coefficient of the $x^{5}$ term is:
$$
C(4,2)=6
$$

Therefore, the coefficient of the $x^{5}$ term is
$$
\frac{1}{8} \times[126+48+4+6]=\frac{1}{8} \times 184=23
$$

The corresponding layout is shown in Figure 4-19.