Question

Two finite sets $A$ and $B$ 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 value of $m$ and $n$ are

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

Answer 1

Answer: (B)

$6,3$

Total number of subsets of set A will be $2^m$
Total number of subsets of set B will be $2^n$
Therefore
$2^m=56+2^n$
$2^m-2^n=2^3\left(7\right)$
$2^n(2^{m-n}-1)=2^3(8-1)$
Now $2^n=2^3$
$n=3$ ...(i)
And $2^{m-n}-1=8-1$
$m-n=3$.
We know that $n=3$
Therefore
$m-3=3$
$m=6$