I am trying to find the number of the longest paths of a complete undirected graph $K_n$. My guess is $\frac{n!}{2}$. Any longest path in $K_n$ should have $n$ vertices. And given one longest path $v_1 - v_2 - \dots - v_n$ by ordering $n$ vertices, the only path which is same with the given one is $v_n - v_{n-1} - \dots - v_1$. Thus, we have $\frac{n!}{2}$.
Thank you in advance!