To calculate the number of possible passwords that meet the given criteria using a set-based approach, follow these steps:

Step 1: Define the set of possible characters.

The characters available are uppercase letters and digits:

• Uppercase letters: 26 characters (A-Z)
• Digits: 10 characters (0-9)

Thus, the total number of available characters is: $26 + 10 = 36$

Step 2: Total passwords (no restrictions).

The total number of passwords of length $n$ (6 to 8 characters) is: $36^n$ This represents all possible combinations of characters for a given password length.

For lengths of 6, 7, and 8, the total is: $\text{Total passwords} = 36^6 + 36^7 + 36^8$

Step 3: Subtract passwords with no digits.

Passwords with no digits consist only of uppercase letters (26 choices per character). The total number of such passwords for a given length $n$ is: $26^n$

For lengths of 6, 7, and 8, the total is: $\text{Passwords with no digits} = 26^6 + 26^7 + 26^8$

Step 4: Subtract the invalid passwords.

The valid passwords are those that contain at least one digit. Using the complement principle: $\text{Valid passwords} = \text{Total passwords} - \text{Passwords with no digits}$

Substituting the values: $\text{Valid passwords} = (36^6 + 36^7 + 36^8) - (26^6 + 26^7 + 26^8)$

Step 5: Perform the calculations.

Let's compute each term:

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

Thus: $36^6 + 36^7 + 36^8 = 2,176,782,336 + 78,364,164,096 + 2,821,109,907,456 = 2,901,650,853,888$

Similarly:

1. $26^6 = 308,915,776$
2. $26^7 = 8,031,810,176$
3. $26^8 = 208,827,064,576$

Thus: $26^6 + 26^7 + 26^8 = 308,915,776 + 8,031,810,176 + 208,827,064,576 = 217,167,790,528$

Finally, subtract: $\text{Valid passwords} = 2,901,650,853,888 - 217,167,790,528 = 2,684,483,063,360$

Final Answer:

The total number of valid passwords is: $\boxed{2,684,483,063,360}$

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Would you like a detailed breakdown of any specific step? Here are related questions for exploration:

1. What would the result be if the password length were fixed at 7 characters?
2. How does the inclusion of lowercase letters affect the calculation?
3. What is the probability that a random password contains exactly two digits?
4. How would the calculation change if special characters (e.g., @, #, $) were included?
5. What is the average number of digits in a valid password?

Tip: Always use the complement principle for "at least one" constraints to simplify combinatorial