To calculate the total number of valid passwords, let's use standard mathematical notation.

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Total possibilities for a password:

1. Passwords are 6 to 8 characters long.
2. Each character is an uppercase letter or a digit, so $26 + 10 = 36$ possible characters per position.

Let $L = 36$ be the number of possible characters.

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Case 1: Password length $n = 6$

Total passwords (with no restrictions):

$L^6 = 36^6$

Passwords with no digits:

If only uppercase letters are used ($26$ choices):
$26^6$

Valid passwords (at least one digit):

$36^6 - 26^6$

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Case 2: Password length $n = 7$

Total passwords:

$L^7 = 36^7$

Passwords with no digits:

$26^7$

Valid passwords:

$36^7 - 26^7$

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Case 3: Password length $n = 8$

Total passwords:

$L^8 = 36^8$

Passwords with no digits:

$26^8$

Valid passwords:

$36^8 - 26^8$

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Total number of valid passwords:

Summing over all cases: $\text{Total valid passwords} = (36^6 - 26^6) + (36^7 - 26^7) + (36^8 - 26^8)$

This expression provides the total number of possible passwords that meet the criteria.

Would you like me to compute this value numerically or provide further insights into this calculation