<p>Let \(A\) and \(B\) be two sets containing 2 elements and 4 elements, respectively. The number of subsets of \(A \times B\) having 3 or more elements is</p>
Step-by-Step Solution
Key Concept: The total number of subsets of A × B is 2^8 (since |A × B| = 2×4 = 8), and we must subtract subsets with 0, 1, or 2 elements using binomial coefficients: C(8,0) + C(8,1) + C(8,2).
<p><strong>Step 1:</strong> Find the cardinality of A × B. Since |A| = 2 and |B| = 4, we have |A × B| = 2 × 4 = 8.</p><p><strong>Step 2:</strong> The total number of subsets of A × B is 2^8 = 256.</p><p><strong>Step 3:</strong> Use complementary counting. Subsets with 3 or more elements = Total subsets - Subsets with 0, 1, or 2 elements.</p><p><strong>Step 4:</strong> Calculate subsets with ≤ 2 elements:<br/>• Subsets with 0 elements: C(8,0) = 1<br/>• Subsets with 1 element: C(8,1) = 8<br/>• Subsets with 2 elements: C(8,2) = 28<br/>• Total: 1 + 8 + 28 = 37</p><p><strong>Step 5:</strong> Subsets with 3 or more elements = 256 - 37 = 219.</p><p>∴ Answer: B</p>
Correct Answer: B