Let A and B be two non-empty sets having n elements in common. Then, the number of elements common to $A×B$ and $B×A$ is

$n^2$

We know that

$(A × B)∩(C × D) = (A∩C) × (B∩D)$

$∴(A × B)∩(B × A) = (A∩B) × (B∩A)$

$⇒(A × B)∩(B × A) = (A∩B) × (A∩B)$

It is given that $A∩B$ has n elements.

$∴(A∩B) × (A∩B)$ has $n^2$ elements.

But, $(A× B)∩(B × A) = (A∩B) × (A∩B)$

$∴(A× B)∩(B × A)$ has $n^2$ elements in common.