\(2n^2\)
To solve the problem, we need to determine the number of elements in the product set \((X - Y) \times (Y - X)\), where \(X\) and \(Y\) are given sets with specific properties:
- Set \(X\) contains \(3n\) elements.
- Set \(Y\) contains \(2n\) elements.
- They have \(n\) elements in common.
First, let's determine the elements in each set that are not in the other:
- \(X - Y\): The elements in \(X\) that are not in \(Y\).
- \(Y - X\): The elements in \(Y\) that are not in \(X\).
Since \(X\) and \(Y\) have \(n\) elements in common, we can calculate the number of elements in each of these differences:
- \(X - Y\) contains \(3n - n = 2n\) elements.
- \(Y - X\) contains \(2n - n = n\) elements.
Next, we find the Cartesian product \((X - Y) \times (Y - X)\). This product consists of ordered pairs \((a, b)\) where \(a\) is an element of \(X - Y\) and \(b\) is an element of \(Y - X\).
The number of such ordered pairs is equal to the product of the number of elements in \(X - Y\) and \(Y - X\):
\(| X - Y | \times | Y - X | = 2n \times n = 2n^2\)
Therefore, the number of elements in the set \((X - Y) \times (Y - X)\) is \(2n^2\).
Thus, the correct answer is:
\(2n^2\).
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