$36 \cdot 21$ In a room, there are $N$ people $(N>3)$, and at least one person has not shaken hands with everyone in the room. The maximum number of people in the room who could have shaken hands with everyone is
(A) 0.
(B) 1.
(C) $N-1$.
(D) $N$.
(E) None of these.
(29th American High School Mathematics Examination, 1978)
[Solution] Label these people as $A_{1}, A_{2}, \cdots, A_{V}$, and the numbering should be such that $A_{1}$ and $A_{2}$ have not shaken hands.
Perhaps every other pair has shaken hands, but at least $A_{1}$ and $A_{2}$ have not shaken hands with everyone. Therefore, at most $N-2$ people have shaken hands with everyone.
Hence, the answer is $(E)$.