Question:
Two finite sets have m and n elements respectively. The total number of subsets of first set is 56 more than the total number of subsets of the second set. The value of m and n is respectively are:
(a) 7, 6
(b) 5, 1
(c) 6, 3
(d) 8, 7
Solution:
Let us suppose two finite sets are A and B
Let A has m elements
Let B has n elements
Then total number of subjects of A is 2m and total number of subjects of B is 2n.
According to given condition,
2m – 2n = 56
i.e 2n (2m – n – 1) = 56
Since 56 = 8 × 7
$=2^{3} \times 7$
i.e. $2^{n}\left(2^{m-n}-1\right)=2^{3} \times 7$
i.e $n=3$ and $2^{m-n}-1=7$
i.e $2^{m-n}=8$
$2^{m-n}=2^{3}$
i.e. m – n = 3
i.e m = n + 3
m = 6
i.e m = 6, n = 3
Hence, the correct answer is option C.