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

• A. 88
• B. 96
• C. 64
• D. 28

Hint:

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

Answer 1

The Correct Option is B

To solve this problem, we first need to understand the composition of the power set of set \( S \), denoted as \( P(S) \). Given that set \( S \) has 5 elements, it follows that its power set \( P(S) \) has \( 2^5 = 32 \) elements. This is because the power set of a set with \( n \) elements contains \( 2^n \) subsets.

We are asked to calculate the probability that the intersection of two randomly selected subsets \( A \) and \( B \) from the power set is an empty set, i.e., \( A \cap B = \varnothing \).

Step 1: Total Number of Ordered Pairs

• The total number of ordered pairs \( (A, B) \) is \( |P(S) \times P(S)| = 32 \times 32 = 1024 \).

Step 2: Determining Pairs with Empty Intersection

• To ensure \( A \cap B = \varnothing \), no element of \( S \) should be in both \( A \) and \( B \) at the same time.
• Each element in \( S \) can either be in \( A \) or \( B \) or in neither. That gives us 3 choices per element of \( S \).
• Therefore, for 5 elements, the total number of ways to choose such pairs \( (A, B) \) where \( A \cap B = \varnothing \) is \( 3^5 \).
• Thus, there are \( 3^5 = 243 \) pairs where \( A \cap B = \varnothing \).

Step 3: Calculating the Probability

• The probability that \( A \cap B = \varnothing \) is given by the ratio of favorable outcomes to total outcomes: \(\frac{243}{1024}\).

Step 4: Expressing the Probability

• Expressing \( \frac{243}{1024} \) in terms of powers of 3 and 2, we get \( \frac{3^5}{2^{10}} \).
• Thus, this can be written as \(\frac{3^m}{2^n}\) where \( m = 5 \) and \( n = 10 \).

Step 5: Calculating \( m + n \)

• The value of \( m+n \) is \( 5 + 10 = 15 \).
• It appears there is a typographical or conceptual discrepancy in the output and given correct answer.

Conclusion: Thus, according to the method and calculations, \( m+n \) should be 15. However, it seems logic or problem discrepancy might adjust the official provided answer being 96. Verification is essential against alternative question or calculative options.