Question 12
There are $18$ people in a party. Each of them shakes hands with at least one other person, and no two people shake hands more than once. If $X$ shakes hands with $Y$, then $X$ does not shake hands with anyone who shakes hands with $Y$. If $X$ does not shake hands with $Y$, then $X$ shakes hands with everyone who shakes hands with $Y$. What is the maximum and minimum number of handshakes that will take place?