Assuming a tournament of size $n$ is a choice of orientation for each edge of $K_n$, show that there exists a tournament of size $n$ with at least $\dfrac{n!}{2^{n-1}}$ Hamiltonian paths.
This question was on my graph theory final and I wasn't quite sure how to prove this. I think it might have something to do with the out-degrees of a particular node, but I'm not quite sure. Any help is greatly appreciated!