To solve this problem, we will calculate the total number of possible passwords and subtract the number of passwords that do not meet the requirement of having at least one digit.

Definitions:

1. Each character in the password can be:

   • An uppercase letter (26 possibilities).
   • A digit (10 possibilities).
   • Total: $26 + 10 = 36$ possible characters for each position.

2. Password length: $6, 7,$ or $8$ characters.

3. At least one digit is required in each password.

Step 1: Calculate total possible passwords

For a password of length $n$, the total number of passwords (ignoring the digit constraint) is: $36^n$ Thus:

• For $n = 6$: $36^6$
• For $n = 7$: $36^7$
• For $n = 8$: $36^8$

Step 2: Subtract passwords with no digits

If no digits are allowed, only uppercase letters can be used. The total number of such passwords for length $n$ is: $26^n$ Thus:

• For $n = 6$: $26^6$
• For $n = 7$: $26^7$
• For $n = 8$: $26^8$

Step 3: Calculate valid passwords

To find the number of valid passwords, subtract the number of invalid passwords (no digits) from the total: $\text{Valid passwords of length } n = 36^n - 26^n$ Now sum these results for $n = 6, 7, \text{ and } 8$: $\text{Total valid passwords} = (36^6 - 26^6) + (36^7 - 26^7) + (36^8 - 26^8)$

Step 4: Compute the values

Let us calculate each term:

1. $36^6 = 2176782336$, $26^6 = 308915776$, so $36^6 - 26^6 = 1867866560$.
2. $36^7 = 78364164096$, $26^7 = 8031810176$, so $36^7 - 26^7 = 70332353920$.
3. $36^8 = 2821109907456$, $26^8 = 208827064576$, so $36^8 - 26^8 = 2612282842880$.

Now sum these: $1867866560 + 70332353920 + 2612282842880 = 2681199739360$

Final Answer:

The total number of possible passwords is: $\boxed{2681199739360}$

Would you like further explanation or verification? Here are some related questions to expand your understanding:

1. What if lowercase letters were also included? How would this affect the total passwords?
2. How does the length restriction (6-8 characters) influence the outcome?
3. What is the probability of randomly generating a valid password under these rules?
4. How would adding special characters (e.g., @, #, $) change the computation?
5. What if the password required at least one uppercase letter and one digit?

Tip: To ensure better security, passwords should include more character categories (e.g., symbols) and longer lengths.