Question

Two finite sets have $m$ and $n$ element respectively The total number of subsets of the first set is $112$ more than the total number of subsets of the second set The value of $m$ and $n$ respectively are

• A. $5,2$
• B. $4,7$
• C. $7,4$
• D. $2,5$

Answer 1

The correct option is C

$7,4$

Explanation for correct option
Given that the total number of subsets of the first set is $112$ more than the total number of subsets of the second set.
$$\begin{gathered} \Rightarrow 2^m-2^n=112 \\ \Rightarrow 2^n\left(2^{m-n}-1\right)=2^4\times 7 \\ \therefore 2^n=2^4\&\left(2^{m-n}-1\right)=7 \end{gathered}$$
We can say that the value of $n=4$
$$\begin{gathered} \Rightarrow \left(2^{m-n}-1\right)=7 \\ \Rightarrow 2^{m-n}=8 \\ \Rightarrow 2^{m-4}=2^3\mathbf{}\mathbf{[}\mathbf{\because }\mathit{n}\mathbf{=}\mathbf{4}\mathbf{]} \\ \Rightarrow m-4=3 \\ \Rightarrow m=7 \end{gathered}$$
Therefore the value of $m=7\&n=4$

Hence option(C) is correct