How many Hamiltonian cycles are there in a complete graph $K_n$ ($n\geq 3$) Why? It seems to me like the answer should be $n!$. Since you can first select the first vertex, then select each remaining vertex until you've got them all in a permutation.

My textbook, however, says that the answer is $(1/2)(n-1)!$

Unfortunately, it doesn't explain why this is. Why is it?
The answer really depends on how we think that two Hamiltonian cycles are equal. Take $K_3$ as example. Let's call its vertices $1$, $2$ and $3$. Then do we consider $1\to2\to3\to1$ same as $2\to3\to1\to 2$? If yes,  $1\to2\to3\to1$ is the same as $2\to3\to1\to 2$, which is the same as $3\to1\to2\to 3$ in $K_3$. Then we will have two Hamiltonian cycles $1\to 2\to3\to1$ and $1\to 3\to 2\to1$. Moreover, if we consider $1\to 2\to3\to1$ and $1\to 3\to 2\to1$ being the same because the second one is obtained by reversing direction the first one, then we have only one Hamiltonian cycle in $K_3$.

For general $K_n$, it's the same. $1\to2\to\cdots\to n\to1$ is the same as $2\to\cdots\to n\to1\to 2$ is the same as.... $n\to1\to 2\to\cdots \to n$. And the $1\to 2\to\cdots\to n\to1$ and $1\to n\to\cdots\to2\to1$ being the same because the second one is obtained by reversing direction the first one. So we have altogether $\frac{1}{2}(n-1)!$ Hamiltonian cycles in $K_n$.
Easiest way to see this is to draw all possible Hamiltonians as figures - fairly easy to do for K4 say. You will then notice that of the 8 drawn, some are actually duplicated.. there are only 3.

I guess if you did this for K5 etc you would get a pattern to which 1/2(n-1)! is the answer