I can see why you would think that. For n=5 (say a,b,c,d,e) there are in fact n! unique permutations of those letters.
However, the number of cycles of a graph is different from the number of permutations in a string, because of duplicates -- there are many different permutations that generate the same identical cycle.
There are two forms of duplicates:
First, in a cycle there is no starting and ending place. That means that permutation a-b-c-d-e would generate the same cycle as b-c-d-e-a, c-d-e-a-b, d-e-a-b-c and e-a-b-c-d. Notice that there are n duplicate tours in the above, that's where the (n-1)! instead of n! comes from.
Second, in a cycle, direction doesn't matter. That means that permutations a-b-c-d-e and e-d-c-b-a would generate the same cycle. For every permutation, there is exactly one permutation that is it's reverse. That's where the /2 comes from.
Thus, (n-1)!/2 unique cycles.