**Standard automobile license plates in a state display 4 digits (from 0-9), followed by 3 letters (from A to Z).**
**How many different standard plates are possible in this system?**
*(Assume repetitions of letters and numbers are allowed.)*

To determine how many different license plates are possible, let's break the problem down into two parts:

1. **Digits (numbers):** The plate contains 4 digits. Each digit can be any number from 0 to 9, which gives us 10 possible choices per digit. Since there are 4 digits, the total number of possible combinations of digits is:

   \[
   10 \times 10 \times 10 \times 10 = 10^4 = 10,000
   \]

2. **Letters:** The plate also contains 3 letters. Each letter can be any letter from A to Z, which gives us 26 possible choices per letter. Since there are 3 letters, the total number of possible combinations of letters is:

   \[
   26 \times 26 \times 26 = 26^3 = 17,576
   \]

Now, to find the total number of different license plates, multiply the total possibilities for the digits by the total possibilities for the letters:

\[
10,000 \times 17,576 = 175,760,000
\]

So, the total number of different standard plates possible in this system is **175,760,000**.

Would you like further clarification or details on any of these steps?

Here are 5 related questions to expand this concept:
1. What if the state restricted the first digit to be non-zero? How many plates would be possible then?
2. How would the number of plates change if repetitions of letters were not allowed?
3. What if the state used 5 digits and 2 letters instead? How many plates would be possible?
4. If one digit and one letter were pre-fixed (e.g., "5A"), how many plates could be created?
5. How would the system change if only specific letters (e.g., excluding vowels) were allowed?

**Tip:** When solving problems involving combinations, break the problem into smaller parts (e.g., digits and letters), and apply the multiplication rule of counting to get the total number of possibilities.

Standard automobile license plates in a state display 4 digits (from 0-9), followed by 3 letters (from A to Z). How many different standard plates are possible in this system? (Assume repetitions of letters and numbers are allowed.)

Learn how to determine the total number of possible combinations for standard automobile license plates with 4 digits (0-9) and 3 letters (A-Z). This problem covers counting principles, including combinatorics and the fundamental principle of counting. Suitable for high school students in Grades 9-12.

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