Example 2. On a plane, there are $n$ lines, among which $k$ lines are pairwise parallel, and $t$ lines pass through the same point $(k+t<n)$, in addition, any two intersect, and any 3 do not share a point. How many regions do they divide the plane into:
The number of intersection points is given by $p_{n}=C_{n}^{2}-C_{k}^{2}-C_{t}^{2}+(t-1)$. According to Theorem 1, the number of regions they divide the plane into is
$$
S_{n}=C_{n}^{2}-C_{1}^{2}-C_{t}^{2}+n+t.
$$

The figure below gives an example F for $n=8, k=4, t=3$. In this case, $S_{8}=C_{8}^{2}-C_{4}^{2}-C_{3}^{2}+8+3=28-6-3+8+3-1=30$.