Discussion
If two sets \(A\) and \(B\) are having 11 elements in common, then the number of elements common to \(A \times B\) and \(B \times A\)
Answer: A
The number of common elements is 121.
An ordered pair (x, y) lies in both \(A \times B\) and \(B \times A\) exactly when x is in A and in B, and y is in B and in A. So
\[ \begin{aligned} &(A \times B) \cap (B \times A) \\ &= (A \cap B) \times (A \cap B) \end{aligned} \]
With \(n(A \cap B) = 11\), this set has \(11 \times 11 = 121\) elements. 22 adds the two counts instead of multiplying them, and 11 forgets that both coordinates range over the common elements.
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