Suppose we have a complete directed graph of 5 vertices with no self-edge.
That is, for any \(i, j = 1, 2, 3, 4, 5\), edge \((v_i, v_j)\) exists iff \(i \neq j\).
How many edges are in the graph?:
What is the length of the longest path not containing a cycle?
How many paths are there that satisfy the previous condition?
If every edge were connected, there would be \(5\cdot 5 = 25\) edges. However, there are no self-edges, so we've over-counted by \(5\) and there are \(20\) total.
\(5\) unique vertices can be connected by \(4\) edges, so a \(5\)th edge is guaranteed to create a cycle.
Visting the \(5\) vertices in any order works. Thus, there are \(5\) possibilities for the start vertex, \(4\) for the second, \(3\) for the third, \(2\) for the fourth, and the final one is whatever's left. This gives \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120\) possible paths.