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

Answer 1

$A$ and $B$ are two sets having $m,n$ elements respectively.
No.of subsets of $A=2^m$
No.of subsets of $B=2^n$
$2^m=2^n+112$
$2^m-2^n=112$
$2^n(2^{m-n}-1)=112$
$2^n(2^{m-n}-1)=2^4(2^3-1)$
$n=4$
$m-4=3\Rightarrow m=7$
$\Rightarrow m\times n=28$