Discussion
Two finite sets A and B have \(m\) and \(n\) elements, respectively. The total number of subsets of A is 48 more than the number of subsets of B. The values of \(m\) and \(n\), respectively, are
Answer: B
The values are \(m = 6\) and \(n = 4\).
A set with \(k\) elements has \(2^{k}\) subsets, so the condition is a difference of two powers of 2:
\[ \begin{aligned} 2^{m} - 2^{n} &= 48 \\ 2^{n}(2^{m-n} - 1) &= 16 \times 3 \end{aligned} \]
The bracket is odd, so \(2^{n} = 16\) and \(2^{m-n} - 1 = 3\), giving \(n = 4\) and \(m - n = 2\), so \(m = 6\). Check: \(64 - 16 = 48\).
The other pairs fail: 6,3 gives \(64 - 8 = 56\); 5,6 and 2,6 make A the smaller set, so the difference is negative; 7,1 gives \(128 - 2 = 126\).
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