* 6. Among $n(\geqslant 3)$ lines, exactly $p(\geqslant 2)$ are parallel, and no three lines among the $n$ lines intersect at the same point. Then the number of regions into which these $n$ lines divide the plane is
6. $\frac{1}{2}\left(n^{2}-p^{2}+n+p+2\right) \quad p$ parallel lines divide the plane into $p+1$ regions. When adding $r$ of the remaining $n-p$ lines, let the plane be divided into $b_{p+r}$ regions. If another line is added, this line will intersect with the others at $p+r$ points, thus being divided into $p+r+1$ line segments or rays, each of which increases the number of regions by 1, so,
$$
\begin{array}{l}
b_{p+r+1}=b_{p+r}+(p+r+1), \\
\sum_{r=0}^{n-p-1}\left(b_{p+r+1}-b_{p+r}\right)=\sum_{r=0}^{n-p-1}(p+r+1), \\
b_{n}-b_{p}=\frac{1}{2}(n+p+1)(n-p) .
\end{array}
$$

Substituting $b_{p}=p+1$ for $\lambda$. The result is obtained.