Sets — If two sets A and B have n common elements, how many ordered pairs are common to the Cartesian products A × B and B × A?

Introduction / Context:
The intersection (A × B) ∩ (B × A) contains those ordered pairs that appear in both products. Because order matters, a pair (x, y) from A × B appears in B × A as well only in very specific circumstances. Understanding this condition is the key.

Given Data / Assumptions:

• A and B share exactly n elements in common (i.e., |A ∩ B| = n).
• Cartesian products consider ordered pairs: first coordinate from the first set, second from the second set.

Concept / Approach:
(x, y) ∈ A × B requires x ∈ A and y ∈ B. For the same ordered pair (x, y) to also belong to B × A, we must simultaneously have x ∈ B and y ∈ A. Therefore x and y must both lie in A ∩ B, and we additionally need (x, y) = (y, x) to be the same ordered pair in both sets. That equality of ordered pairs forces x = y.

Step-by-Step Solution:
Condition 1: x ∈ A ∩ B and y ∈ A ∩ BCondition 2: (x, y) = (y, x) ⇒ x = yThus common pairs are exactly {(x, x) | x ∈ A ∩ B}Count = |A ∩ B| = n

Verification / Alternative check:
Try a small example: A = {1, 2}, B = {2, 3}. A ∩ B = {2}. A × B contains (2, 2); B × A contains (2, 2) as well; no other pair overlaps. Exactly one common pair equals n = 1.

Why Other Options Are Wrong:
n^2 would count all (x, y) with x, y in A ∩ B, but only diagonal pairs (x, x) are shared; n^3 and 2n are unrelated counts; “None of these” is false because the exact count equals n.

Common Pitfalls:
Ignoring order in ordered pairs; or assuming that if x and y are both common elements, then (x, y) is automatically common between the products — it is not, unless x = y.

Final Answer:
n