# 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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