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. m = 7, n = 6
• B. m = 6, n = 3
• C. m = 5, n = 1
• D. m = 8, n = 7

Answer 1

Answer: (B)

m = 6, n = 3

Answer 2

Let A and B be such sets, i.e, n (A) = m and n (B) = n So, number of subsets of $A = 2^m$ Number of subsets of $B = 2^n$ According to the question, $2^m - 2^n = 56$ $\Rightarrow \, 2^n (2^{m - n} - 1) = 56 = 2^3.7$ Thus, $n = 3$ and $2^{m - n} - 1 = 7$ $\Rightarrow \, n = 3$ and $m - n = 3$ Hence, we get n = 3 and m = 6