The problem asks for the number of possible passwords that meet the following conditions:

1. The password is 10 characters long.
2. The first character must be an uppercase letter (A-Z).
3. The last character must be a single digit (0-9).
4. The remaining 8 characters can be any letter (A-Z, a-z) or digit (0-9).

Step-by-step breakdown:

1. First character: There are 26 possible uppercase letters (A-Z).
2. Last character: There are 10 possible digits (0-9).
3. Middle 8 characters: Each of these can be a letter (uppercase or lowercase) or a digit, giving 26 (uppercase) + 26 (lowercase) + 10 (digits) = 62 possible choices for each of the 8 characters.

Total number of possible passwords:

The total number of possible passwords is the product of the possibilities for each position:

$26 \times 62^8 \times 10$

Let me calculate this for you.The total number of possible passwords is 56,768,427,452,072,960.

Would you like a more detailed explanation or have any further questions?

Here are some related questions for practice:

1. What if the first character could be any letter (uppercase or lowercase)?
2. What if the password length was increased to 12 characters under the same conditions?
3. How many passwords would be possible if the middle characters could only be digits?
4. What is the effect on the total number of passwords if the last character can also be a letter (A-Z, a-z)?
5. How many passwords are possible if there are no restrictions on the characters?

Tip: When counting possibilities in a sequence, multiply the options for each position to get the total number of combinations.