<p>Let <span>X = \{1, 2, 3, \ldots, 12\}</span> and <span>N</span> be the number of pairs <span>\{A, B\}</span> such that <span>A ⊆ X</span>, <span>B ⊆ X</span>, <span>A ≠ B</span> and <span>A ∩ B = \{2, 3, 5, 7, 8\}</span>. Then the value of <span>N</span> is</p>
Step-by-Step Solution
Key Concept: Given that A ∩ B must equal a fixed set {2, 3, 5, 7, 8}, both A and B must contain these 5 elements, and no other common elements. The remaining elements can be distributed among A only, B only, or neither to count valid pairs where A ≠ B.
<p><strong>Step 1: Identify the fixed elements.</strong></p><p>Since A ∩ B = {2, 3, 5, 7, 8}, both A and B must contain all 5 elements {2, 3, 5, 7, 8}. The remaining 7 elements are X \ {2, 3, 5, 7, 8} = {1, 4, 6, 9, 10, 11, 12}.</p><p><strong>Step 2: Classify the remaining elements.</strong></p><p>For each of the 7 remaining elements, we have 4 choices:</p><ul><li>Element in A only</li><li>Element in B only</li><li>Element in neither A nor B</li><li>Element in both A and B (but this contradicts A ∩ B = {2, 3, 5, 7, 8})</li></ul><p>So each of the 7 remaining elements has exactly 3 valid choices (not in both A and B simultaneously).</p><p><strong>Step 3: Count total ordered pairs (A, B).</strong></p><p>Each of the 7 elements can be assigned to one of 3 categories: 'in A only', 'in B only', or 'in neither'. This gives 3^7 ordered pairs (A, B).</p><p><strong>Step 4: Handle the constraint A ≠ B.</strong></p><p>We must subtract the case where A = B. For A = B with A ∩ B = {2, 3, 5, 7, 8}, we need A = B = {2, 3, 5, 7, 8} (since the 7 remaining elements cannot be in both A and B simultaneously for A ∩ B to equal exactly {2, 3, 5, 7, 8}). There is exactly 1 such pair.</p><p><strong>Step 5: Account for unordered pairs.</strong></p><p>The problem asks for pairs {A, B}, which are unordered. The number of ordered pairs (A, B) with A ≠ B is 3^7 - 1 = 2187 - 1 = 2186. The number of unordered pairs is:</p><p>N = (3^7 - 1)/2 = 2186/2 = 1093</p><p><strong>∴ Answer: 1093</strong></p>
Correct Answer: 1093