1. **Problem Statement:** We are given two sets $p$ and $q$ with $|p| = 10$ and $|q| = 15$. We need to find the number of elements in the Cartesian products $p \times q$, $q \times p$, and $p \times p$. 2. **Formula Used:** The number of elements in the Cartesian product of two sets $A$ and $B$ is given by: $$|A \times B| = |A| \times |B|$$ This means the size of the product set is the product of the sizes of the individual sets. 3. **Calculations:** - For $p \times q$: $$|p \times q| = |p| \times |q| = 10 \times 15 = 150$$ - For $q \times p$: $$|q \times p| = |q| \times |p| = 15 \times 10 = 150$$ - For $p \times p$: $$|p \times p| = |p| \times |p| = 10 \times 10 = 100$$ 4. **Explanation:** The Cartesian product $A \times B$ consists of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$. Since each element of $A$ pairs with every element of $B$, the total number of pairs is the product of the sizes of $A$ and $B$. **Final answers:** - Number of elements in $p \times q$ is 150. - Number of elements in $q \times p$ is 150. - Number of elements in $p \times p$ is 100.