To solve this problem, let's break it down based on the Fundamental Counting Principle.

Problem Analysis

The license plate in question has two components:

1. Letters: There are 2 letters.
2. Numbers: There are 3 numbers.

The problem also states that repetitions are allowed, which means each letter or number can be used more than once.

Step-by-Step Solution

1. Letters: There are 26 letters in the English alphabet, so each letter slot has 26 possible choices.

   • For the first letter: 26 options.
   • For the second letter: 26 options.

   Total combinations for the letter portion = $26 \times 26 = 676$.

2. Numbers: Each number can be from 0 to 9, providing 10 possible choices per number slot.

   • For the first number: 10 options.
   • For the second number: 10 options.
   • For the third number: 10 options.

   Total combinations for the number portion = $10 \times 10 \times 10 = 1000$.

3. Combine the Two Parts: By the Fundamental Counting Principle, the total number of unique license plates is the product of the combinations of the two parts. $676 \times 1000 = 676000$

Answer

The total number of different license plates that can be manufactured is: $\boxed{676000}$

Would you like more details or have any questions?

Related Questions:

1. What if there were 3 letters and 3 numbers in the license plate?
2. How would the solution change if repetitions were not allowed?
3. What if each letter and number position had a specific set of allowable characters or digits?
4. How many license plates can be formed if only consonants are used for the letters?
5. How would the total change if additional characters (like hyphens) were allowed?

Tip: When using the Fundamental Counting Principle, always count the choices for each part separately and multiply them.