How can we calculate the total number of paths joining any pair of points in a collection of $N$ points?
Specifically, consider the following example of a set of four points (their distribution on a square is not important) :
Now we could join any pair of these 4 points like these (I don't think I missed some paths):
For three points only, there are 6 paths.
Crossing of lines is permited. A path shouldn't come back to the start point. For a set of 4 points, I found 30 paths possible. But what should be the formula for a set of $N$ points ?
So, you are picking $k-$vertices out of $n$ and then forming a path on them, so you have to give them an ordering in which the path is gonna pass. Notice that this creates a symmetry (do i go on the path forwards or backwards). The formula, then, should be $$\frac{1}{2}\sum _{k=2}^n\binom{n}{k}k!$$