I give up, how many?
How many mate in 6s are there from the starting position?
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More than 12
There are none, as sdet said, because "mate in X" means that one player can force mate in X moves with best play by the other player, and that is not possible from the starting position. Maybe you mean "how many ways are there to achieve a mate on move 6 from the starting position?" There is a very large number of those, due to the number of different irrelevant moves both sides can do to collaborate in achieving mate on move 6.
How many mate in 6s are there from the starting position?