 Standard automobile license plates in a country (using English Alphabet) display 4 numbers, followed by 3 letters.

      If repetition of letters and numbers is ALLOWED , then there are  175760000 possible different standard plates in this system.


      If repetition of letters and numbers is NOT ALLOWED , then there are  __________ possible different standard plates in this system.


We can solve this step-by-step by first understanding the problem.

### Case: No repetition of numbers or letters
The total number of possible different standard plates is composed of 4 numbers followed by 3 letters.

1. **For the numbers (no repetition):**
   - There are 10 digits (0-9), and we need to choose 4 numbers.
   - The first digit has 10 options, the second digit has 9 options (since repetition is not allowed), the third has 8, and the fourth has 7.

   Thus, the number of ways to choose 4 numbers is:
   \[
   10 \times 9 \times 8 \times 7 = 5040
   \]

2. **For the letters (no repetition):**
   - There are 26 letters (A-Z), and we need to choose 3 letters.
   - The first letter has 26 options, the second has 25, and the third has 24.

   Thus, the number of ways to choose 3 letters is:
   \[
   26 \times 25 \times 24 = 15600
   \]

### Total number of possible plates (no repetition):
To find the total number of possible plates without repetition, we multiply the possibilities for the numbers and the letters:
\[
5040 \times 15600 = 78624000
\]

Thus, the number of possible different standard plates when repetition is **not allowed** is **78,624,000**.

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Would you like any more details or have questions on this? Here are some related questions:

1. How does the total number of plates change if there were only 2 letters and 4 numbers?
2. What if the repetition was allowed for letters but not for numbers?
3. How many plates are possible if only the letters are restricted from repetition?
4. How would the calculation change if the letters could include special characters or numbers?
5. What if the license plates required 5 digits and 2 letters, still without repetition?

**Tip:** When repetition is not allowed, always consider the reducing number of available options for each subsequent choice!

Standard automobile license plates in a country (using English Alphabet) display 4 numbers, followed by 3 letters. If repetition of letters and numbers is ALLOWED, then there are 175760000 possible different standard plates in this system. If repetition of letters and numbers is NOT ALLOWED, then there are __________ possible different standard plates in this system.

Discover how to calculate the number of possible standard license plates using 4 numbers and 3 letters when repetition is not allowed. This solution involves key combinatorics concepts and permutation formulas. Perfect for students in Grades 9-12, learn how to compute the total of 78,624,000 possible plates.

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