Example 4-9 Embed 3 red beads and 2 blue beads on a ring (as shown in Figure 4-11), how many different schemes are there.


$$
\begin{array}{l}
G:(1)(2)(3)(4)(5),(12345),(13524) \\
\quad(14253),(15432) \\
\quad(1)(25)(34),(2)(13)(45) \\
\quad(3)(15)(34),(4)(12)(35)
\end{array}
$$
$$
\text { (5) }(14)(23) \text {. }
$$

Corresponding to $g_{0}=(1)(2)(3)(4)(5)$, there are $C(5,3)=10$ schemes with 3 red and 2 blue tiles, which remain unchanged under $g_{0}$. For (12345) and the other 4 schemes of (5) format, the number of schemes is 0. For the $(1)^{1}(2)^{2}$ format, the number of schemes is 2, hence
$$
N=\frac{1}{10} \times[10+5 \times 2]=2
$$