10 i believe, mite be a few more that are just easter eggs, that can be entered into the vault before day 6 but im not certain.
Exactly, the game does have easter eggs, but according to the actual storyline, I would say that there is 1 password for each character that you wish to save, so we have 1 for Benson, Ty, Dean etc.....
MILD GAMEPLAY SPOILER*
You need to find 2 passwords in order to save Benson, remember that saving Benson is the key of being able to move forward in some of the currently avaible routes, such like Ty's and Dean's. The first password will allow certain conversations to take place, where a keyword will be repeated, that's the one that will allow you to save Benson.
Again... you don't actually need all of the passwords. Just focus on the character whose route want to play. Like Ty.... YES I FUCKING LOVE TY LOL.