1978 AHSME Problems/Problem 16

Problem

In a room containing $N$ people, $N > 3$ , at least one person has not shaken hands with everyone else in the room. What is the maximum number of people in the room that could have shaken hands with everyone else?

$\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }N-1\qquad \textbf{(D) }N\qquad \textbf{(E) }\text{none of these}$

Solution

We assume that one person hasn't shaken hands with the other N people, meaning that they only had shaken hands with N-1 people. However, this doesn't make sense! There will be 1 person ( from the N-1) people who hasn't shaken hands with the first person, meaning that they also only had shaken hands with N-1 people. Therefore, there is a minimum of 2 people in the room that haven't shaken hands with one person. Therefore, the maximum is N-2 people, so the answer is $E$

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1978 AHSME Problems/Problem 16

Problem

Solution