Suppose Silas, Suyeon and Calvin are \(3\) close friends in a class of \(50\) students. There is to be a field trip to a local science museum, but the bus can only take \(25\) students. The (rather devilish) teacher devises an unusual sorting procedure; she randomly divides the \(50\) students into \(25\) pairs, and then each pair flips a coin to see which member of the pair gets to go on the field trip.

The probability that the \(3\) close friends mentioned above all end up going on the field trip is \(\dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a + b\).

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