To solve the problem, we need to calculate the number of possible passwords with the constraints provided:

• Passwords are 6, 7, or 8 characters long.
• Each character can either be an uppercase letter (26 options) or a digit (10 options).
• Passwords must contain at least one digit.

Step-by-Step Solution

1. Total number of passwords (unrestricted):

For a password of length $n$, the total number of possible combinations is $36^n$ because there are 36 possible characters (26 letters + 10 digits).

For lengths 6, 7, and 8: $36^6 + 36^7 + 36^8$

2. Passwords with no digits:

If there are no digits, all characters must be uppercase letters. For length $n$, the number of such passwords is $26^n$.

For lengths 6, 7, and 8: $26^6 + 26^7 + 26^8$

3. Subtract passwords with no digits from the total:

The total number of valid passwords (with at least one digit) is: $\text{Valid passwords} = (36^6 + 36^7 + 36^8) - (26^6 + 26^7 + 26^8)$

Calculate the Values:

1. $36^6 = 2,176,782,336$
2. $36^7 = 78,364,164,096$
3. $36^8 = 2,821,109,907,456$

Adding these together: $36^6 + 36^7 + 36^8 = 2,901,650,854,888$

4. $26^6 = 308,915,776$
5. $26^7 = 8,031,810,176$
6. $26^8 = 208,827,064,576$

Adding these together: $26^6 + 26^7 + 26^8 = 217,167,790,528$

Finally: $\text{Valid passwords} = 2,901,650,854,888 - 217,167,790,528 = 2,684,483,064,360$

Final Answer:

The total number of valid passwords is: $\boxed{2,684,483,064,360}$

Would you like me to break this further or clarify any specific part?

Here are five follow-up questions:

1. How would the answer change if lowercase letters were allowed?
2. What happens if there is a minimum number of digits (e.g., at least 2 digits)?
3. How does the calculation change if the password length is restricted to 8 characters only?
4. What if the password could also include special characters (e.g., symbols like @, #)?
5. How would you model this problem programmatically in Python?

Tip: Always use complement counting (like we did here with "no digits") for problems with "at least one" constraints—it simplifies the math!