At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
All of the handshakes will involve at least one person from the $10$ who knows no one.  Label these ten people $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$, $J$. Person $A$ from the group of $10$ will initiate a handshake with everyone else ($29$ people).  Person $B$ initiates $28$ handshakes plus the one already counted from person $A$.  Person $C$ initiates $27$ new handshakes plus the two we already counted.  This continues until person $J$ initiates $20$ handshakes plus the nine we already counted from $A$ ... $I$. $29+28+27+26+25+24+23+22+21+20 = \fbox{245}$