Note: kindly fill the box with how choice we have in all different cases while explaining it briefly.

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 start with a or b, it also must contain at most three lowercase-letters (including the first letter) and contains no repeated digits (letters can be repeated). How many valid passwords are there? You are required to show your work step-by-step.