Where to sit?

A class of 30 students contains 15 boys and 15 girls. Their classroom has 15 desks, and exactly 2 students can sit at each desk.

At the start of lessons one morning, the 30 students go into their classroom and seat themselves randomly, so that all possible arrangements of pupils are equally likely. The expected number of girls who end up sitting at a desk with another girl can be written as \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?

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