Expected number of elements in an intersection

Let \(P_{n}\) be the set of all subsets of the set \([n] = \{1,2,\ldots, n\}.\) If two distinct elements of \(P_{5}\) are chosen at random, the expected number of elements (of \([n]\)) that they have in common can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b?\)

Details and assumptions

As an explicit example, \( P_2 = \left\{ \emptyset, \{1\}, \{2\}, \{1, 2\} \right\} \). \( \{1\} \) and \(\{1,2\} \) are 2 elements of \(P_2\), which have 1 element in common.

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