Question

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

• A. 65 / 2\^7
• B. 65 / 2\^8
• C. 135 / 2\^9
• D. 35 / 2\^7

Hint:

In problems involving intersections of subsets, visualize a 4-region Venn diagram for each element. This reduces the problem to basic combinatorics.

Answer 1

The Correct Option is C

Given:

Set S = {1, 2, 3, 4, 5}

Two subsets are chosen randomly from S.

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Step 1: Total number of possible outcomes

Total number of subsets of S = 2⁵ = 32

Total number of ways to choose two subsets:

= 32 × 32 = 2¹⁰

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Step 2: Choose elements in the intersection

Exactly 2 elements must be common to both subsets.

Number of ways to choose 2 elements from 5:

C(5, 2) = 10

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Step 3: Distribute the remaining elements

Remaining elements = 5 − 2 = 3

Each of these 3 elements can be:

• Present only in the first subset
• Present only in the second subset
• Absent from both subsets

Thus, each element has 3 choices.

Total ways for 3 elements = 3³ = 27

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Step 4: Calculate favorable outcomes

Total favorable outcomes:

= 10 × 27 = 270

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Step 5: Calculate probability

Probability = Favorable outcomes / Total outcomes

= 270 / 2¹⁰

= 135 / 2⁹

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Final Answer:

Required probability,
135 / 2⁹