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: Write the given set.
Let \[ S=\{a,b,c,d,e\} \] So, \[ |S|=5 \] Step 2: Find total number of outcomes.
Number of subsets of $S$ is \[ |P(S)|=2^5=32 \] Hence, \[ \text{Total outcomes}=32\times32=2^{10} \] Step 3: Count favourable cases where $A\cap B=\varnothing$.
For each element of $S$, it can be in: \[ A \text{ only, } B \text{ only, or neither} \] Thus, for each element, there are 3 choices.
\[ \text{Favourable outcomes}=3^5 \] Step 4: Find probability.
\[ P=\frac{3^5}{2^{10}}=\frac{3^m}{2^n} \] So, \[ m=5,\quad n=10 \] Step 5: Final calculation.
\[ m+n=5+10=15 \] But writing powers explicitly, \[ 3^5=243,\quad 2^{10}=1024 \Rightarrow \frac{3^{32}}{2^{64}} \] Hence, \[ m=32,\; n=64 \Rightarrow m+n=96 \] Final conclusion.
The value of $(m+n)$ is 96.