<Question>
How many license plates can be formed if every license plate has 2 different letters (A through Z) followed by 2 different one digit (0-9) numbers?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out how many different license plates can be made where each plate has 2 different letters followed by 2 different digits. Let me start by breaking down the problem step by step.

<prompt_1.1> Break down the problem into parts to determine the number of combinations for letters and digits separately. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, the license plate structure is: Letter, Letter, Digit, Digit. But wait, the problem says "2 different letters" and "2 different one digits". So the letters must be distinct from each other, and the digits must also be distinct. That’s important. So repetition isn't allowed within the letters or the digits.

Let me start with the letters. There are 26 letters in the English alphabet, right? For the first letter, I can choose any of the 26. But since they have to be different, for the second letter, I can't repeat the first one. So that would be 25 choices. So for the letters part, it's 26 options for the first letter and 25 for the second. So 26 multiplied by 25. Let me write that as 26 × 25.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, moving on to the digits. There are 10 digits (0-9). Similarly, the first digit can be any of the 10, and the second digit has to be different, so 9 choices.
So that's 10 × 9 for the digits part.

To get the total number of license plates, I think I need to multiply the number of possibilities for the letters by the number of possibilities for the digits. Because for each combination of letters, there are all the possible combinations of digits. So total combinations would be (26 × 25) × (10 × 9).

Let me compute that step by step.

<prompt_2.1> Compute the total number of license plates by multiplying the combinations for letters and digits. </prompt_2.1>
-----

Wait, but let me make sure I didn't miss anything. Is there any restriction on the letters or digits being in a certain order?

<prompt_2.2> Re-evaluate the problem to ensure no restrictions are missed and confirm the calculation. </prompt_2.2>

But let me confirm with another approach. For permutations, the number of ways to choose 2 distinct letters from 26 is given by the permutation formula P(n, k) = n! / (n - k)! .

<prompt_2.3> Use permutations to verify the calculation for letters and digits. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
First, 26 × 25. Let me calculate that: 26 × 25. Hmm, 25 times 20 is 500, and 25 times 6 is 150, so 500 + 150 = 650. So 26 × 25 = 650.

Then, 10 × 9 = 90.

Now multiply those two results together: 650 × 90. Let's break that down. 600 × 90 = 54,000 and 50 × 90 = 4,500. Adding those together: 54,000 + 4,500 = 58,500. So 650 × 90 = 58,500.
</execution_2.1>

<execution_2.2>
The problem just says "2 different letters" and "2 different one digit numbers". So I think the order matters here. For example, AB would be different from BA, and similarly for the digits. So permutations, not combinations. Because if it were combinations, the order wouldn't matter. But in license plates, the order does matter. For example, AB12 is different from BA21.

So yes, permutations are the right approach here. So my initial calculation is correct.
</execution_2.2>

<execution_2.3>
So for letters, that's P(26, 2) = 26! / (26 - 2)! = 26 × 25. Which matches what I did before. Similarly, for digits, P(10, 2) = 10 × 9. So that's correct.

Therefore, multiplying them together gives the total number of license plates: 26 × 25 × 10 × 9.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, let me check that multiplication again. 650 × 90.

<prompt_3.1> Re-verify the multiplication for the total number of license plates. </prompt_3.1>
-----

Alternatively, I can think of it as two separate permutation problems. The first is permuting 26 letters taken 2 at a time, and the second is permuting 10 digits taken 2 at a time. Then the total number of license plates is the product of these two permutations.

<prompt_3.2> Confirm the solution using different approaches and finalize the answer. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Another way: 650 × 90 = 650 × (100 - 10) = 650×100 - 650×10 = 65,000 - 6,500 = 58,500. Yep, that's the same. So 58,500 total license plates.
</execution_3.1>

<execution_3.2>
So, P(26, 2) × P(10, 2) = (26 × 25) × (10 × 9) = 650 × 90 = 58,500. Yep, same result.

Is there a chance I might have made a mistake in the multiplication? Let me verify once more. 26 × 25: 25 × 20 is 500, 25 × 6 is 150, so 500 + 150 = 650. Then 10 × 9 is 90. Then 650 × 90. Let's think of 650 × 9 × 10. 650 × 9: 600 × 9 = 5,400 and 50 × 9 = 450, so 5,400 + 450 = 5,850. Then multiply by 10: 5,850 × 10 = 58,500. Correct.

Alternatively, 650 × 90 is the same as 65 × 9 × 100. 65 × 9: 60×9=540, 5×9=45, so 540+45=585. Then 585 × 100 = 58,500. Yep.

I don't see any mistakes here. So the total number of possible license plates is 58,500.
</execution_3.2>
</Execution_3>

<Final_answer>
Therefore, the total number of license plates that can be formed is $\boxed{58,\!
500}$.
</Final_answer>
</think>