No, if you multiply by colors it gets much worse. 100's of thousands of possibilities...
Of course not all parts exist in all colors so that helps (a bit), but it is quite an interesting problem to work on. Every assumption you make will be challenged.
Prints and stickers are counted separately but that's not a really huge number and they should be correctly identified (so the surface decoration matters as well in the classification).
That's a really hard problem. In many cases there are subtle hints (color, the writing on the studs).
Spectrography might give a hint here due to the different formulation of the plastics but some of the knock-offs are now so good it can be very hard to tell them from the real thing.
I'm not really sure if 'counterfeit' is the right term, the companies selling these are not making pieces labeled 'lego', and in fact the Lego brand started out by copying an English product.
It might not be true at the piece level, but some sets on some websites(ali, etc.) are exact replicas of lego sets, bar the brand. Down to the manual. Hence the use of "counterfeit".
Of course not all parts exist in all colors so that helps (a bit), but it is quite an interesting problem to work on. Every assumption you make will be challenged.
Prints and stickers are counted separately but that's not a really huge number and they should be correctly identified (so the surface decoration matters as well in the classification).