Question

Two finite sets have $m$ and $n$ elements respectively. The total number of subsets of the first set is $48$ more than the total number of subsets of the second set and $f=I-\left\{N\right\}$ then $f(m+n)$ is

• A. $9$
• B. $10$
• C. $48$
• D. $15$

Answer 1

The correct option is B

$10$

Find the sum of subsets:

The number of the subset of the first set is $48$ more than that of the second subset,

$2^m=2^n+48$

$$\begin{gathered} \Rightarrow 2^m-2^n=48 \\ \Rightarrow 2^n(2^{m-n}-1)=2^4\times 3 \end{gathered}$$

Now,

$2^n=2^4$ and $2^{m-n}-1=3$

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

Now, $2^{m-n}-1=3$

$$\begin{gathered} \Rightarrow 2^{m-n}=4 \\ \Rightarrow 2^{m-n}=2^2 \\ \Rightarrow m-n=2.......\left(2\right) \end{gathered}$$

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

$m=6$

Now, $m+n=6+4=10$

Hence, the correct option is (B).