With 18 total princesses it would, in theory, be 18*17*16 for a total of 4896. And I think that applies if order of princesses doesnt matter, ie if Oni, Ghost, Bird gives 1 code and Bird, Oni, Ghost gives a different one.
If that isnt the case and either order doesn't give different codes or order is set then I think it brings it down to a potential of about 1632.
Well since I just had a dupe group in my attempt to get as many codes as possible. I now believe the groups to be set unless I just hit the, at best, .03-ish% chance of getting dupe groups.