<p>Let <i>A</i> and <i>B</i> be two sets containing four and two elements respectively. Then the number of subsets of the set <i>A</i> × <i>B</i>, each having at least three elements is:</p>
Step-by-Step Solution
Key Concept: The cardinality of \(A \times B\) is 8. Use complementary counting: total subsets minus those with fewer than 3 elements.
<p><strong>Concept:</strong> Find subsets of \(A \times B\) with at least 3 elements.</p><p><strong>Step 1:</strong> \(|A| = 4\), \(|B| = 2\), so \(|A \times B| = 4 \times 2 = 8\)</p><p><strong>Step 2:</strong> Total number of subsets of \(A \times B\) is \(2^8 = 256\)</p><p><strong>Step 3:</strong> Subsets with at least 3 elements = Total subsets − (Subsets with 0, 1, or 2 elements)</p><p><strong>Step 4:</strong> Subsets with 0 elements: \(\binom{8}{0} = 1\)</p><p>Subsets with 1 element: \(\binom{8}{1} = 8\)</p><p>Subsets with 2 elements: \(\binom{8}{2} = 28\)</p><p><strong>Step 5:</strong> Subsets with 0, 1, or 2 elements: \(1 + 8 + 28 = 37\)</p><p><strong>Step 6:</strong> Subsets with at least 3 elements: \(256 - 37 = 219\)</p><p>∴ Answer is B (219).</p>
Correct Answer: D