Why Can't We Be Friends

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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 \(a+b\).

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