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

Answer 1

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 2^m and total number of subjects of B is 2ⁿ.
According to given condition,
2^m – 2ⁿ = 56
i.e 2ⁿ (2^{m – n} – 1) = 56
Since 56 = 8 × 7

= 2³ × 7

i.e. 2ⁿ (2^{m – n} – 1) = 2³ × 7
i.e n = 3 and 2^{m – n} – 1 = 7
i.e 2^{m – n} = 8
2^{m – n} = 2³
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.