TITLE: Passwords: Two 50-characters vs one 100-characters
QUESTION [25 upvotes]: In this Information Security question, we discuss whether or not a $100$ character secret randomly-generated username is equivalent to a $50$ character secret randomly-generated username plus a $50$ character secret randomly-generated password.
This answer [now deleted] claims that there is a mathematical difference.
It claims:

If we assume that the user id can be kept private and is choosen randomly, it would allow for more combinations. If we make an example with a base of $62$ possible characters to choose from $(a..z, A..Z, 0-1)$, we get:
$62^{100} = 10^{179}$ combinations [versus] $62^{50} + 62^{50} = 80^{89}$ combinations

Is this correct?
It seems erroneous to me; requiring two $50$ character items would be the same number of combinations as requiring one $100$ character item.  If I'm mistaken, can you help me understand my error?

REPLY [2 votes]: In the context of services which allow online registration (the vast majority of existing websites), there is indeed a difference. It's trivial to check the validity of the username alone by simply trying to register a new user with that name. For valid usernames, the site will have no option but to report that the name is already taken.
In case of a brute-force attack (trying every possible combination) guessing a valid username then a valid passowrd leads to the $62^{50} + 62^{50}$ result cited in the answer.