The question asks:

If two sets $A$ and $B$ have 10 elements in common between them, find the number of elements common in the sets $A \times B$ and $B \times A$.

Let’s break down the solution.

1. Understanding Cartesian Products:

   • $A \times B$ represents the Cartesian product of sets $A$ and $B$. It consists of ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.
   • Similarly, $B \times A$ consists of ordered pairs $(b, a)$ where $b \in B$ and $a \in A$.

2. Elements in Common Between $A \times B$ and $B \times A$:

   • The only way an element from $A \times B$ will match an element in $B \times A$ is if it is of the form $(a, a)$, where $a$ is in both $A$ and $B$.
   • Since $A$ and $B$ have 10 elements in common, these 10 elements form pairs $(a, a)$ in both $A \times B$ and $B \times A$.

3. Result:

   • Therefore, there will be exactly 10 such pairs $(a, a)$ that are common in both $A \times B$ and $B \times A$.

Final Answer:

The number of elements common in the sets $A \times B$ and $B \times A$ is 10.

Would you like a detailed explanation of Cartesian products or related examples?

Here are some related questions you might explore:

1. What is the general formula for the number of elements in a Cartesian product $A \times B$ if $A$ and $B$ are finite sets?
2. How would the solution change if $A$ and $B$ had different numbers of common elements?
3. Can there be elements common to $A \times B$ and $B \times A$ if $A$ and $B$ have no elements in common?
4. What is the difference between $A \times B$ and $B \times A$ in terms of ordered pairs?
5. How would this problem extend if there were three sets $A$, $B$, and $C$ with shared elements?

Tip: In set theory, elements in the Cartesian product $A \times B$ depend heavily on the order of the sets, so reversing the order typically changes the set composition unless it’s a set of identical pairs like $(a, a)$.