Question

Let \( A \) and \( B \) be two finite sets with \( m \) and \( n \) elements respectively. The total number of subsets of the set \( A \) is 56 more than the total number of subsets of \( B \). Then the distance of the point \( P(m, n) \) from the point \( Q(-2, -3) \) is:

• A. 10
• B. 6
• C. 4
• D. 8

Answer 1

The Correct Option is A

The total number of subsets of a set with \(m\) elements is \(2^m\) and for a set with \(n\) elements is \(2^n\). Given:  
\(2^m = 2^n + 56.\)

Rearranging:  
\(2^m - 2^n = 56.\)

Factoring the left side:  
\(2^n (2^{m-n} - 1) = 56.\)

Since \(56 = 2^3 \times 7\), we set \(2^n = 8 \implies n = 3\) and  
\(2^{m-n} - 1 = 7 \implies 2^{m-n} = 8 \implies m - n = 3.\)
Therefore:  
\(m = 6, \quad n = 3.\)

The distance between points \(P(6, 3)\) and \(Q(-2, -3)\) is given by:

\(\text{Distance} = \sqrt{(6 - (-2))^2 + (3 - (-3))^2} = \sqrt{8^2 + 6^2} = \sqrt{100} = 10.\)

Thus, the correct answer is 10.