Explanation:

To understand why the correct answer is option 'D', let's break down the problem step by step.

Step 1: Let's define the sets A and B. Assume that A has m elements and B has k elements.

Step 2: The cartesian product of A and B, denoted as (A x B), is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. Since A has m elements and B has k elements, the cardinality (number of elements) of (A x B) can be calculated as m * k.

Step 3: Now, we need to find the number of elements that (A x B) and (B x A) have in common. To do this, we need to find the set of ordered pairs that are common to both (A x B) and (B x A).

Step 4: Let's consider an ordered pair (a, b) that is common to both (A x B) and (B x A). This means that a is an element of A and b is an element of B, and also b is an element of A and a is an element of B. In other words, a is an element of both A and B, and similarly, b is an element of both A and B.

Step 5: Since sets A and B have n elements in common, there are n elements that satisfy the conditions in Step 4. Therefore, the number of elements that (A x B) and (B x A) have in common is equal to the number of elements that satisfy the conditions in Step 4, which is n.

Conclusion: Thus, the correct answer is option 'D', which states that (A x B) and (B x A) will have n^2 (n squared) elements in common.