Question

Set $A$ has $m$ elements and set $B$ has $n$ elements. If the total number of subsets of $A$ Is $112$ more than the total number of subsets of $B$ then the value of $m\times n$ Is:

• A. $25$
• B. $22$
• C. $28$
• D. $24$

Answer 1

The correct option is C

$28$

Finding the value of $m\times n$:

Given, set $A$ has $m$ elements and set $B$ has $n$ elements

No. of subsets of $A=2^m$

No. of subsets of $B=2^n$

Also, total number of subsets of $A$ Is $112$ More than the total number of subsets of $B$.

Therefore, we can write:

$$\begin{gathered} \Rightarrow 2^m=2^n+112 \\ \Rightarrow 2^m-2^n=112 \\ \Rightarrow 2^n(2^{m-n}-1)=112 \\ \Rightarrow 2^n(2^{m-n}-1)=2^4(2^3-1) \\ \Rightarrow n=4\&m-n=3 \\ \Rightarrow m-4=3 \\ \Rightarrow m=7 \\ \therefore m\times n=28 \end{gathered}$$

Hence, the correct answer is option (C).