Question

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

• A. $10,8$
• B. $6,3$
• C. $8,6$
• D. $9,5$

Answer 1

The correct option is B $6,3$

Total number of subsets of the first is $2^m$ while total number of subsets of the second is $2^n$.

Now it is given that

$2^m=2^n+56$

$2^m-2^n=56$

$2^3(2^{m-3}-2^{n-3})=56$

$2^{m-3}-2^{n-3}=7$

$2^{m-3}-2^{n-3}=8-1$

$2^{m-3}-2^{n-3}=2^3-2^0$

Hence comparison gives us

$m-3=3$ thus $m=6$ and

$n-3=0$ giving $n=3$.

Hence, $(m,n)=(6,3)$.