Example 1-25 5 boys, 3 girls sit around a circular table. If there are no additional requirements, then $Q(8,8)=7!=5040$. If it is required that boy $B_{1}$ does not sit next to girl $G_{1}$, how many arrangements are there? If it is required that the 3 girls do not sit next to each other, how many arrangements are there?
Method 1 First, exclude $G_{1}$, and have 7 people sit around a round table, then insert $G_{1}$, as shown in Figure 1-10. $G_{1}$ has 5 choices for insertion, so the number of arrangements where $B_{1}$ and $G_{1}$ are not adjacent is $5 \times 6! = 5 \times 720 = 3600$.

Method 2 First, find the number of arrangements where $B_{1}$ and $G_{1}$ sit adjacent: $2 \times 6! = 2 \times 720 = 1440$, $5040 - 1440 = 3600$.

If the 3 girls are not adjacent. First, consider the number of ways 5 boys can sit around a round table, then insert the 3 girls one by one, the number of arrangements is
$$
5! \times 5 \times 4 \times 3 = 1440
$$