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Two random subsets of a set: probability of a given intersection

This type of question, asked 2 times: 2021 Paper 2, 24 Jan 2026 S2. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    64
    The probability that two randomly selected subsets of the set {1,2,3,4,5}\mathbf{\{ 1,2,3,4,5\}} have exactly two elements in their intersection is

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  2. Question 2

    24 Jan 2026 S2

    74
    Let S be a set of 5 elements and P(S)P(S) denote the power set of S . Let E be an event of choosing an ordered pair (A, B) from the set P(S)×P(S)P(S) \times P(S) such that A∩B=∅A \cap B = \varnothing. If the probability of the event E is 3p2q\frac{3^{p}}{2^{q}}, where p,q∈Np,q \in \mathbb{N}, then p+qp + q is equal to