TITLE: Number of unique paths in a complete graph with n verticies
QUESTION [5 upvotes]: Sorry if this question has been answered before, but I am having difficulty finding it in the suggested list of questions on graph theory.
Consider a complete graph with $n$ vertices and $\frac{n(n-1)}{2}$ edges. Is there an equation which describes the number of unique paths through such a graph from any one vertex to another, that is the number of unique sets of vertices where the start point is connected to the end point without re-intersecting the start?  
For example, I guess for a complete graph with three vertices there would be just two ways to get from point A to point B without re-intersecting the starting point.

REPLY [8 votes]: You must mean non-self-intersecting paths, otherwise the answer would be infinite.  So: given a starting vertex $a$, and an ending vertex $b$, you can choose any subset of $k$ of the remaining $n-2$ vertices, and use them as intermediate vertices of a path in whatever order you like.  So the answer is
$$ \sum_{k=0}^{n-2} \binom{n-2}{k}k!,$$
which is (an offset of) OEIS sequence A000522: $1, 1, 2, 5, 16, 65, \ldots$.