I'm stuck on a rather peculiar question (for lack of an advanced math background), that boils down to this: is it possible to quantify the probability of finding n points in space, in a certain configuration? Configuration here implies rotational invariance, that is, if I have coordinates of all n points, they don't move with respect to each other if they are fixed into a certain configuration.
For example, say you have 8 points defining the vertices of a cube. If you rotate the cube in 3D, the configuration I am talking of doesn't change. The coordinates however, do change. What is the probability of finding those 8 points in such a configuration, out of all possible configurations that they could possibly be in?