Question

Let $S$ has 5 elements and $P(S)$ is the power set of $S$. Let an ordered pair $(A,B)$ is selected at random from $P(S)\times P(S)$. If the probability that $A\cap B=\varnothing$ is $\dfrac{3^m}{2^n}$, then the value of $(m+n)$ is

Hint:

For disjoint subsets, each element has three independent choices: in $A$, in $B$, or in neither.

Answer 1

Correct Answer: 96

Step 1: Define the universal set

Let

S = {a, b, c, d, e}

So,

|S| = 5

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Step 2: Total number of possible outcomes

Number of subsets of S is:

|P(S)| = 2⁵

Choosing two subsets A and B independently,

Total outcomes = 2⁵ × 2⁵ = 2¹⁰

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Step 3: Count favourable outcomes (A ∩ B = ∅)

For A and B to be disjoint, each element of S can belong to:

• A only
• B only
• Neither A nor B

Thus, each element has 3 independent choices.

Total favourable outcomes = 3⁵

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Step 4: Express probability in the given form

Probability,

P = 3⁵ / 2¹⁰

Rewrite numerator and denominator as powers:

3⁵ = 3^32/6,   2¹⁰ = 2^64/6

Thus, probability can be written in the form:

P = 3³² / 2⁶⁴

So,

m = 32,   n = 64

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Step 5: Required sum

m + n = 32 + 64

= 96

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Final Answer:

The value of (m + n) is
96