Discussion
Let A and B be two sets of having 3 and 2 elements respectively. Then the number of subsets of \(A \times B\) having at least three elements is
Answer: C
There are 42 such subsets.
\(A \times B\) has \(3 \times 2 = 6\) elements, so it has \(2^{6} = 64\) subsets in all. Remove those with fewer than three elements:
\[ \begin{aligned} \text{fewer than 3} &= \binom{6}{0} + \binom{6}{1} + \binom{6}{2} \\ &= 1 + 6 + 15 = 22 \\ \text{at least 3} &= 64 - 22 = 42 \end{aligned} \]
64 is the total number of subsets, and 52 and 60 are too large: they leave in some of the 22 subsets with fewer than three elements.
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