$35 \cdot 32$ A disk is divided by $2 n$ equally spaced radii $(n>0)$ and a chord. The maximum number of non-overlapping regions into which the disk can be divided is
(A) $2 n+1$.
(B) $2 n+2$.
(C) $3 n-1$.
(D) $3 n$.
(E) $3 n+1$.
(22nd American High School Mathematics Examination, 1971)
[Solution] For $2 n$ equal sectors $a_{1}, a_{2}, \cdots, a_{2 n}$, a cutting line $P Q$ can only cut across $(n+1)$ sectors, dividing each into two parts.
The total number of different regions is
$$
2 n+(n+1)=3 n+1 \text {. }
$$

Therefore, the answer is $(E)$.