A computer password is required to be 9 characters long. How many passwords are possible if the password requires 1 letter(s) followed by 8 digits (numbers 0-9), where no repetition of any letter or digit is allowed?

There are  possible passwords.

To solve this problem, we need to determine how many possible passwords can be created under the given constraints:

1. **One letter** followed by **eight digits**, with no repetition of any letter or digit.

### Step-by-step approach:

1. **Choosing the letter**:  
   The password starts with one letter. Since the alphabet has 26 letters (A-Z), there are 26 possible choices for the first character.

2. **Choosing the first digit**:  
   The second character (first digit) must be one of the digits from 0 to 9, so there are 10 possible choices for this character.

3. **Choosing the second digit**:  
   Since there is no repetition allowed, the second digit must be chosen from the remaining 9 digits (as one digit is already used). So, there are 9 choices for the third character.

4. **Choosing the third digit**:  
   Similarly, the next digit must be selected from the remaining 8 digits, so there are 8 possible choices for the fourth character.

5. **Proceeding in this manner**:  
   The number of choices for the subsequent digits will decrease by 1 each time:
   - For the fourth character, there will be 7 choices.
   - For the fifth character, there will be 6 choices.
   - For the sixth character, there will be 5 choices.
   - For the seventh character, there will be 4 choices.
   - For the eighth character, there will be 3 choices.
   - For the ninth character, there will be 2 choices.

### Total number of possible passwords:
We multiply the number of choices at each step:

\[
26 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2
\]

Let me calculate this.The total number of possible passwords is \( 94,348,800 \).

Would you like more details on this solution or have any questions? 

Here are 5 related questions:
1. How would the solution change if repetition of digits was allowed?
2. What if the password required a mix of uppercase and lowercase letters?
3. How would the number of passwords change if the password length increased to 10 characters?
4. What is the probability of guessing the correct password on the first try?
5. How does the number of possible passwords compare if the password had no restrictions?

**Tip:** For combinatorial problems, always break the task down step by step, accounting for each condition.

A computer password is required to be 9 characters long. How many passwords are possible if the password requires 1 letter(s) followed by 8 digits (numbers 0-9), where no repetition of any letter or digit is allowed?

Explore how to calculate the number of possible passwords with one letter followed by eight digits, with no repetition of characters. This solution applies key combinatorics concepts and the principle of counting, making it ideal for high school students in Grades 10-12.

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