We are given two sets, $A$ and $B$, and the problem asks us to find the number of elements in the Cartesian product $A \times B$.

Step-by-step Solution:

1. Set $A$: $A = \{0, 1, 2, \dots, 8\}$ The elements in $A$ are from 0 to 8, inclusive. The number of elements in $A$ is: $|A| = 9 \quad (\text{since there are 9 elements from 0 to 8})$

2. Set $B$: $B = \{2, 3, 4, \dots, 26\}$ The elements in $B$ are from 2 to 26, inclusive. The number of elements in $B$ is: $|B| = 26 - 2 + 1 = 25$ There are 25 elements in set $B$.

3. Cartesian Product $A \times B$: The Cartesian product $A \times B$ consists of all ordered pairs $(a, b)$, where $a \in A$ and $b \in B$. The number of elements in the Cartesian product is the product of the number of elements in $A$ and $B$: $|A \times B| = |A| \times |B| = 9 \times 25 = 225$

Final Answer:

The number of elements in $A \times B$ is $\boxed{225}$.

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Would you like further details or clarifications?

Here are 5 related questions to explore:

1. What is the general formula for the number of elements in the Cartesian product of two sets?
2. How would the result change if set $B$ had fewer elements, say from 2 to 20?
3. What is the significance of Cartesian products in coordinate geometry?
4. Can Cartesian products be used to represent real-world relationships? If yes, how?
5. How do Cartesian products behave with infinite sets?

Tip: The Cartesian product of two sets gives you a way to pair elements, which is crucial in defining relations and functions in mathematics.