Question

Two finite sets have $m,n$ elements respectively. The total number of subsets of the first set is $224$ more than the total number of subsets of the second. The values of $m$ and $n$ are

• A. $5,7$
• B. $4,8$
• C. $8,5$
• D. $8,7$

Answer 1

The correct option is C $8,5$

As number of elements in $A$ and $B$ are $m,n$ respectively
$\therefore$ Number of subsets of $A=2^m$
Number ofsubsets of $B=2^n$
According to the problem,
$2^m-2^n=224=32\times 7$
$2^n(2^{m-n}-1)=2^5\times (8-1)$

$\Rightarrow 2^n(2^{m-n}-1)=2^5\times (2^{8-5}-1)$

$\Rightarrow n=5,m-n=8-5$
$\Rightarrow n=5,m=3+5=8$