Question

Two finite sets have $m$ and $n$elements. The total number of subsets of the first is $56$ more than the total number of subsets of the second set. The values of $m$ and $n$are

• A. $7,6$
• B. $6,3$
• C. $5,1$
• D. $8,7$

Answer 1

The correct option is B

$6,3$

Find the required values :

Let set $A$ have $m$ elements and set $B$ have $n$elements

Now, $2^m-2^n=56$

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

On comparing both sides

$2^n=2^3$ and $2^{m-n}-1=7$

Now, $n=3.....\left(1\right)$

So, $2^{m-n}-1=7$

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

From $\left(1\right)$ and $\left(2\right)$

$m=6$

Hence, the correct option is (B).