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

• A. $7,6$
• B. $6,3$
• C. $5,1$
• D. $8,7$

Answer 1

Answer: (B)

No. of subsets if a set contain $r$ elements $=2^{r}$
$\therefore 2^{m}-2^{n}=56 $
$\Rightarrow 2^{n}\left(2^{m-n}-1\right)=8 \times 7=2^{3} \times 7$
$\therefore n=3 \text { and } 2^{m-n}=8=2^{3}$
$\Rightarrow m-n=3 $
$\Rightarrow m-3=3 $
$\Rightarrow m=6 $
$\therefore m=6, n=3$