Question

Two finite sets have $m$ and $n$ elements. The total number of subsets of the first set is $56$ more than the total number of subsets of the second set. The values of $m$ and $n$ are respectively.

• A. $8,5$
• B. $6,3$
• C. $4,1$
• D. None of these

Answer 1

The correct option is B $6,3$

Number of subsets of the set having $m$ element is given by $2^m$.

Thus number of subsets of the set having $m$ element is $2^m$ and that having $n$ element is $2^n$.

According to the question, $2^m=2^n+56$
$\Rightarrow 2^m-2^n=56$

$\Rightarrow 2^n(2^{m-n}-1)=56$
$\Rightarrow 2^n(2^{m-n}-1)=8(8-1)=2^2(2^3-1)$

$\Rightarrow n=3$ and $m-n=3$
$\Rightarrow m=n+3=3+3=6$.