This might be a duplicate (sorry), but I can't seem to figure out a neat closed form for this.
Question: Suppose I have a regular $n$-gon and I want to pick exactly $k$ pairs of adjacent edges from it, where $k < [n/2]$. How many different ways are there to do this?
Obviously, if $k = n/2$ and $n$ is even, then there are only 2 ways to do this. But if $k < n/2$ then it starts to get complicated. It starts to depend on the distance between pairs.
For example, if $n=6$ and $k=2$, then the pairs could themselves be adjacent (i.e. make up a path of length 4) and there are 6 different combinations of possible 4-paths. But the pairs could also be opposite each other (i.e. the pairs have an edge between them on either side). Then there are also 3 ways to arrange those for a total of 6.