<p>We know that \(A \times B\) will have eight elements. Out of these 8 elements, the total number of subsets containing 3 or more elements is \({}^8C_3 + {}^8C_4 + {}^8C_5 + {}^8C_6 + {}^8C_7 + {}^8C_8 = 2^8 - {}^8C_0 - {}^8C_1 - {}^8C_2\). The total number of subsets containing 3 or more elements is:</p>
Step-by-Step Solution
Key Concept: Use the binomial theorem identity: total subsets = 2^n, then subtract subsets with fewer than 3 elements (using C(n,0) + C(n,1) + C(n,2)) to find subsets with 3 or more elements.
<p><strong>Step 1:</strong> Recognize that the total number of subsets of a set with 8 elements is 2<sup>8</sup> = 256.</p><p><strong>Step 2:</strong> Subsets with fewer than 3 elements are those with 0, 1, or 2 elements: <sup>8</sup>C₀ + <sup>8</sup>C₁ + <sup>8</sup>C₂ = 1 + 8 + 28 = 37.</p><p><strong>Step 3:</strong> By complementary counting, subsets with 3 or more elements = 2<sup>8</sup> - (<sup>8</sup>C₀ + <sup>8</sup>C₁ + <sup>8</sup>C₂) = 256 - 37 = 219.</p><p><strong>Step 4:</strong> Verify: <sup>8</sup>C₃ + <sup>8</sup>C₄ + <sup>8</sup>C₅ + <sup>8</sup>C₆ + <sup>8</sup>C₇ + <sup>8</sup>C₈ = 56 + 70 + 56 + 28 + 8 + 1 = 219 ✓</p><p>∴ Answer: <strong>219</strong></p>
Correct Answer: B