Let A and B be two sets containing four and two elements respectively. The…

Concept

If a finite set has n elements, its total number of subsets is 2n. The number of subsets having exactly r elements is the binomial coefficient C(n, r). To count subsets with “at least k” elements, it is easier to subtract the small cases from the total: subsets with at least k elements = 2n − (number of subsets with fewer than k elements).

Application

1. Find the size of A × B. Since |A| = 4 and |B| = 2, the Cartesian product has |A × B| = 4 × 2 = 8 ordered pairs.

2. Total number of subsets of an 8-element set is 28 = 256.

3. Count the subsets to exclude — those with fewer than three elements (0, 1 or 2 elements): C(8, 0) + C(8, 1) + C(8, 2) = 1 + 8 + 28 = 37.

4. Subtract: subsets with at least three elements = 256 − 37 = 219.

Cross-check

Add the wanted cases directly instead of subtracting: Σ from r = 3 to 8 of C(8, r) = 56 + 70 + 56 + 28 + 8 + 1 = 219. Both routes agree, confirming the count of 219.