Question

Two finite sets have $ m $ and $ n $ number of elements respectively. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Then the values of $ m $ and $ n $ are respectively.

• A. \( 7, 4 \)
• B. \( 7, 7 \)
• C. \( 4, 4 \)
• D. \( 4, 7 \)

Hint:

To solve such problems, express the total number of subsets as powers of 2, then use the given difference to form an equation and solve for \( m \) and \( n \).

Answer 1

The Correct Option is A

The total number of subsets of a set with \( n \) elements is given by \( 2^n \). Let the number of elements in the first set be \( m \) and in the second set be \( n \). Thus, the number of subsets of the first set is \( 2^m \), and the number of subsets of the second set is \( 2^n \). According to the given condition, we have the equation: \[ 2^m = 2^n + 112 \] Now, we need to find the values of \( m \) and \( n \) that satisfy this equation. First, test for \( m = 7 \) and \( n = 4 \): \[ 2^7 = 128, \quad 2^4 = 16 \] Substitute these into the equation: \[ 128 = 16 + 112 \] This is true, so the values of \( m \) and \( n \) are \( 7 \) and \( 4 \), respectively. Thus, the correct answer is \( (A) \).