A state makes license plates with three letters followed by three numbers with no repetitions of the letters permitted. How many different possibilities are there?

The problem asks how many different possibilities exist for a license plate that consists of three letters followed by three numbers, with the restriction that no letters can be repeated. However, the numbers can be repeated.

We will break the problem down step by step:

1. **Choosing the three letters**:  
   The alphabet contains 26 letters. Since no repetition of letters is allowed:
   - For the first letter, there are 26 options.
   - For the second letter, there are 25 remaining options (since one letter is already used).
   - For the third letter, there are 24 remaining options (since two letters are already used).

   The total number of ways to select the letters is the product of these choices:
   \[
   26 \times 25 \times 24
   \]

2. **Choosing the three numbers**:  
   The digits range from 0 to 9, giving 10 possible digits. Since there are no restrictions on repetition for the numbers:
   - For the first number, there are 10 options.
   - For the second number, there are 10 options.
   - For the third number, there are 10 options.

   The total number of ways to select the numbers is:
   \[
   10 \times 10 \times 10
   \]

3. **Calculating the total number of possibilities**:  
   The total number of different possibilities for the license plates is the product of the number of ways to choose the letters and the number of ways to choose the numbers:
   \[
   (26 \times 25 \times 24) \times (10 \times 10 \times 10)
   \]
   Let's compute this:
   \[
   26 \times 25 \times 24 = 15,600
   \]
   \[
   10 \times 10 \times 10 = 1,000
   \]
   Therefore, the total number of possibilities is:
   \[
   15,600 \times 1,000 = 15,600,000
   \]

So, there are **15,600,000** different possible license plates.

Would you like further clarification on any part of the solution? Here are some related questions to think about:

1. What if repetition of letters was allowed?
2. How would the result change if the numbers couldn't be repeated?
3. How many possibilities would there be if only two letters and four numbers were used?
4. How many possibilities are there if both the letters and numbers are allowed to repeat?
5. What if there was only one number and five letters instead?

**Tip:** When working with problems involving choices or combinations, think about whether each choice is dependent or independent, and whether repetition is allowed!

A state makes license plates with three letters followed by three numbers with no repetitions of the letters permitted. How many different possibilities are there?

Discover how to calculate the number of different possibilities for a state license plate with three unique letters followed by three numbers. This problem involves combinatorial concepts and the fundamental principle of counting, making it suitable for students in Grades 9-12.

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