GPT-3.5 | Conversation #1127765855825039380

[short zinc]

Students are required to create 6-character long passwords to access the library. The
letters must be from lowercase letters or digits. Each password must contain at most two
lowercase-letters and contains no repeated digits. The first letter must be either "a" or
"1". How many valid passwords are there? You are required to show your work step-by
step.

[dire hamlet]

To find the number of valid passwords, we can break down the problem into multiple steps:

Step 1: Determine the possible choices for the first character.
Since the first letter must be either "a" or "1", we have 2 choices for the first character.

Step 2: Determine the possible choices for the second character.
As there should be at most two lowercase letters, we consider the cases with 0, 1, or 2 lowercase letters as the second character.

• Case 1: 0 lowercase letters
  In this case, the second character must be a digit. There are 10 possible choices (0-9).

• Case 2: 1 lowercase letter
  In this case, we have two subcases: when the second character is a lowercase letter or when it is a digit.

  • Subcase 2.1: Second character is a lowercase letter
    Since we can't use the same lowercase letter twice, we have 25 choices for the second character (excluding 'a').

  • Subcase 2.2: Second character is a digit
    We have 9 choices for the second character (excluding '1').

• Case 3: 2 lowercase letters
  Similar to Case 2, we have two subcases: when both characters are lowercase letters or when one is a lowercase letter and the other is a digit.

  • Subcase 3.1: Both characters are lowercase letters
    In this subcase, we have 25 choices for the second character (excluding 'a'), and 24 choices for the third character (excluding the first and second characters).

  • Subcase 3.2: One character is a lowercase letter and the other is a digit
    In this subcase, we have 25 choices for the second character (excluding 'a'), and 9 choices for the third character (excluding '1' and the second character).

Step 3: Determine the possible choices for the remaining characters (fourth, fifth, and sixth characters).
The remaining characters can be any digit (0-9) or lowercase letter (excluding the ones already used).

Now, let's calculate the total number of valid passwords by multiplying the number of choices in each step:

Total