Example 1. On the three sides of $\triangle A B C$, there are $l, m, n$ points respectively. Lines are drawn from each vertex to the points on its opposite side. If no three of these lines intersect at the same point, how many regions do they divide $\triangle A B C$ into?
This problem is relatively simple. First, draw $l$ lines from $A$, obtaining $l+1$ regions. Consider any line drawn from $B$, which is divided into $l+1$ segments by the $l$ lines drawn from $A$, thus increasing the number of regions by $l+1$. If $m$ lines are drawn from $B$, the number of regions increases by $m(l+1)$. Finally, consider any line drawn from $C$, which is divided into $l+m+1$ segments by the $l+m$ lines drawn from $A$ and $B$, increasing the number of regions by $l+m+1$. Thus, drawing $n$ lines from $C$ increases the number of regions by $n(l+m+1)$. Therefore, the number of regions formed by these lines in $\triangle A P C$ is
$$
\begin{aligned}
S_{1} & =l+1+m(l+1)+n(l+m+1) \\
& =l m+m n+n l+l+m+n+1 .
\end{aligned}
$$