## Handshakes

A group of $200$ high school students, $105$ girls and $95$ boys, is randomly divided into two rows of $100$ students each. Each student in one row is directly opposite a student in the other row, and all the opposite pairs of students shake hands. Prove that the number of girl-girl handshakes is $5$ more than the number of boy-boy handshakes.
Source: NCTM Mathematics Teacher 2008

SOLUTION
Let $bb$ be the number of boy-boy handshakes; $gg$ the number of girl-girl handshakes; $bg$ the number of boy-girl handshakes.
$2bb+bg=95$
$2gg+bg=105$
$95-2bb=105-2gg$
$2gg-2bb=10$
$gg-bb=5$
$gg=bb+5$
The number of girl-girl handshakes is $5$ more than the number of boy-boy handshakes.