9! is the total number of possible configurations of the puzzle, whereas 9!/2 is the total number of solvable configurations. For example, this configuration doesn’t have a solution:
1 2 3
4 5 6
8 7
Read more about the solvability of certain configurations of the n-puzzle in this Wikipedia article, or as pointed out by @dasblinkenlight in this MathWorld explanation.
One possible way to find out that 9!/2 is the number of solvable configurations is to start from a solved puzzle and generate all the possible valid, non-repeating movements from it.