What Are Permutations and Factorials?

In a tournament with players $\{x_1, ..., x_n\}$, a Hamiltonian path is a permutation of these players, $\{y_1, ..., y_n\}$ such that $y_1$ beats $y_2,$ $y_2$ beats $y_3,$ and so on.

In a tournament of five players numbered $1$ through $5,$ which of the following pairs of permutations can simultaneously be Hamiltonian paths?