To solve this problem, we need to determine how many different passwords of length six can be formed from the available big letters, small letters, numbers, and special characters, while satisfying the requirement that each password contains at least one big letter, one number, or one special character.

Total number of passwords without any restrictions:

1. The total number of possible characters for each position in the password is:

   • 29 big letters
   • 29 small letters
   • 10 numbers
   • 34 special characters

   The total number of characters available is: $29 + 29 + 10 + 34 = 102$

2. Since there are no restrictions at first, and each of the six positions in the password can be filled with any of the 102 characters, the total number of unrestricted passwords is: $102^6$

Total number of passwords with no big letters, no numbers, and no special characters:

To find the number of passwords that fail to meet the requirement, we count those consisting entirely of small letters (since passwords made purely of small letters violate all conditions).

• There are 29 small letters available, so the number of invalid passwords (those made entirely of small letters) is: $29^6$

Valid passwords:

To find the number of valid passwords, subtract the number of invalid passwords from the total number of possible passwords: $102^6 - 29^6$

Let's compute the values now.

1. $102^6$
2. $29^6$

After performing the calculations, we can subtract the two results to obtain the final answer.The total number of valid passwords, where the password contains at least one big letter, one number, or one special character, is 1,125,567,595,943.

Would you like more details or have any questions?

Here are five related questions:

1. How would the answer change if repetition of characters wasn't allowed?
2. What if the password length was increased to 8 characters?
3. How does the solution change if only numbers or special characters are allowed?
4. What if the password must contain at least one of each category (big letter, small letter, number, special character)?
5. How would the number of valid passwords change if only 5 special characters were available?

Tip: For problems involving "at least one" conditions, it often helps to use the complement rule by subtracting invalid cases.