If the number of common elements between A × B and B × A is 25, what is the number of common elements between A and B?

We are given that the number of common elements between $A \times B$ and $B \times A$ is 25. We need to find the number of elements common to $A$ and $B$, i.e., $n(A \cap B)$.

An ordered pair $(x, y)$ is common to $A \times B$ and $B \times A$ if and only if:

• $(x, y) \in A \times B \implies x \in A \text{ and } y \in B$
• $(x, y) \in B \times A \implies x \in B \text{ and } y \in A$ For $(x, y)$ to be in both sets, we must have $x \in A$ and $x \in B$, which means $x \in A \cap B$. Similarly, $y \in B$ and $y \in A$, which means $y \in A \cap B$. Therefore, the set of common elements is $(A \cap B) \times (A \cap B)$.

Let $n(A \cap B) = k$.
The number of elements in $(A \cap B) \times (A \cap B)$ is $n((A \cap B) \times (A \cap B)) = n(A \cap B) \cdot n(A \cap B) = k \cdot k = k^2$.
We are given that this number is 25.

Taking the square root of both sides:

Since $k$ represents the number of elements, it must be a non-negative integer.

Thus, the number of elements common to $A$ and $B$ is 5.

The calculated value is 5, which corresponds to option (C).

The final answer is $\text{C}$