Question

Two finite sets $A$ and $B$ have $m$ and $n$ elements, respectively. If the total number of subsets of $A$ is$112$ more than the total number of subsets of $B$, then the value of $m$is

• A. $7$
• B. $9$
• C. $10$
• D. $12$
• E. $13$

Answer 1

The correct option is A

$7$

Find the value of $m$:

Given,

The number of subsets of $A$is $112$ greater than the number of subsets of $B$

Now,

$$\begin{gathered} \Rightarrow 2^m-2^n=112 \\ \Rightarrow 2^n(2^{m-n}-1)=2^4\times 7 \end{gathered}$$

Now, on comparing both sides

$2^n=2^4$and $2^{m-n}-1`=7$

So, $n=4.......\left(1\right)$

$$\begin{gathered} \Rightarrow 2^{m-n}-1=7 \\ \Rightarrow 2^{m-n}=8 \\ \Rightarrow 2^{m-n}=2^3 \\ \Rightarrow m-n=3......\left(2\right) \end{gathered}$$

Now, from $\left(1\right)$ and $\left(2\right)$

$m=7$

Hence, the correct answer is (A).