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

Step 1: Total number of pairs of subsets $(A, B)$ is $2^5 \times 2^5 = 2^{10}$.
Step 2: For each element $x \in \{1, 2, 3, 4, 5\}$, there are 4 cases: $x \in A \cap B$, $x \in A \cap B^c$, $x \in A^c \cap B$, or $x \in A^c \cap B^c$.
Step 3: We want exactly 2 elements in $A \cap B$.
Step 4: Select 2 elements for $A \cap B$ in ${}^5C_2$ ways.
Step 5: For the remaining 3 elements, each has 3 choices (it cannot be in $A \cap B$).
Step 6: Favorable cases $= {}^5C_2 \times 3^3 = 10 \times 27 = 270$.
Step 7: $P = \frac{270}{2^{10}} = \frac{135}{2^9}$.