Solution to P1(a) Using Set and Mathematical Format:

Definitions:

1. Character Set: Uppercase letters (A-Z) and digits (0-9), which totals: $|S| = 26 \, (\text{letters}) + 10 \, (\text{digits}) = 36$

2. Password Lengths: Password length is between 6 and 8 characters, inclusive.

Total Possible Passwords Without Restriction:

For passwords of length $n$, where $n \in \{6, 7, 8\}$: $\text{Total passwords for length } n = |S|^n = 36^n$ Thus, the total unrestricted passwords are: $\text{Total unrestricted passwords } = 36^6 + 36^7 + 36^8$

Invalid Passwords (No Digits):

If a password contains no digits, then all characters must be uppercase letters. The size of this subset is: $|L| = 26 \quad \text{(only letters in the subset)}$ For each password length $n$: $\text{Invalid passwords for length } n = |L|^n = 26^n$ Thus, the total invalid passwords (no digits) are: $\text{Total invalid passwords } = 26^6 + 26^7 + 26^8$

Valid Passwords (At Least One Digit):

Using the complement principle, the valid passwords are: $\text{Valid passwords } = \text{Total unrestricted passwords} - \text{Total invalid passwords}$ Substituting: $\text{Valid passwords } = (36^6 + 36^7 + 36^8) - (26^6 + 26^7 + 26^8)$

Final Calculation:

Expanding the powers:

1. $36^6 = 2176782336$
2. $36^7 = 78364164096$
3. $36^8 = 2821109907456$

$\text{Total unrestricted passwords } = 2176782336 + 78364164096 + 2821109907456 = 2909823384896$

1. $26^6 = 308915776$
2. $26^7 = 8031810176$
3. $26^8 = 208827064576$

$\text{Total invalid passwords } = 308915776 + 8031810176 + 208827064576 = 217948085528$

$\text{Valid passwords } = 2909823384896 - 217948085528 = 2691875299368$

Answer:

$\boxed{2691875299368 \, \text{valid passwords}}$

Would you like me to expand or verify any part of this calculation?

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5 Related Questions:

1. What is the probability of randomly generating a valid password?
2. How does increasing the character set size (e.g., adding symbols) impact the number of valid passwords?
3. What if the password length was 5–8 characters instead? How would that affect the result?
4. How many passwords are valid if at least two digits are required?
5. Can this method be adapted to include lowercase letters and symbols?

Tip: To minimize calculation errors, work incrementally by calculating powers and sums separately before combining results.