Sudoku grids are simply special cases of Latin squares, and the enumeration of Latin squares is itself a difficult problem, with no general combinatorial formulae known…. It is known that the number of 9 by 9 Latin squares is 5524751496156892842531225600. Since this answer is enormous, we need to refine our search considerably in order to be able to get an answer in a sensible amount of computing time.
So write Bertram Felgenhauer of the Department of Computer Science at TU Dresden and Frazer Jarvis Department of Pure Mathematics University of Sheffield in their study “Enumerating possible Sudoku grids.”
But Kjell Fredrik Pettersen explains that “There are 4743933602050718 essentially different Sudoku 2×5 grids.” His study of called, through no mere coincidence, “There are 4743933602050718 essentially different Sudoku 2×5 grids.”
(Thanks to investigators Sari Breck and Tim Yettin, respectively, for bringing these to our attention.)
UPDATE: Investigator Marijke Keet adds:
You might find it of interest that others [2,6] use a count of a mere 6,670,903,752,021,072,963,960 — for solutions for a 9 x 9 grid (see [3,7] for an informal summary of [2,6]) — which has wikipedia as referenced source , that there, in turn, references back to Felgenhauer and Jarvis, but then a 2005 reference in the text, which is actually Jarvis’ homepage that is last updated in 2006 . Either way, both  and, a.o.,  offer some solutions that are more or less scalable for finding solutions, which, presumably, is what most people want to arrive at as opposed to just the possible combinations.