<p>Consider a set \(X = \{1, 2, 3, \ldots, 9, 10\}\). What is the number of pairs \(\{A, B\}\) such that \(A \subseteq X\) and \(B \subseteq X\) also \(A \neq B\) and \(A \cap B = \{2, 3, 5, 7\}\)</p>
Step-by-Step Solution
Key Concept: For each element in X∖{2,3,5,7}, decide independently whether it belongs to A only, B only, both, or neither—then divide by 2 since {A,B} is an unordered pair.
<p><strong>Step 1:</strong> Identify the fixed elements. Since A∩B = {2,3,5,7}, these four elements must be in both A and B.</p><p><strong>Step 2:</strong> For the remaining 6 elements {1,4,6,8,9,10}, each element has 4 independent choices: be in A only, in B only, in both A and B, or in neither.</p><p><strong>Step 3:</strong> This gives 4^6 = 4096 ordered pairs (A,B) satisfying the conditions.</p><p><strong>Step 4:</strong> Since A ≠ B is already enforced (we cannot have all 6 remaining elements in both sets only, as that would make A = B), we need to count unordered pairs.</p><p><strong>Step 5:</strong> The number of unordered pairs {A,B} = (4^6)/2 = 4096/2 = <strong>2048</strong>.</p><p>∴ Answer: A</p>
Correct Answer: A