Why Can't We Be Friends

Ten people $a_n$ , with shirts numbered from $n = 1$ to $10$ are in a meeting room. If the probability that $a_m$ and $a_n$ become friends is

$\frac{m+mn+m}{110}$

the probability that, by choosing two random people, they become friends, can be written as $\frac{a}{b}$ , where $a$ and $b$ are positive coprime numbers. Evaluate $a+b$ .