Containing a Regular Polygon by Extending Sides

We consider the number of ways in which k sides can be selected from a regular n-sided polygon so that, when the chosen sides are extended infinitely on both ends, they form a k-gon that contains the n-gon. We find formulas for three interpretations: the n sides are distinguishable, or rotations are indistinguishable, or both rotations and reflections are indistinguishable.