I'm looking for an analytical solution to a very simple problem I've come across. Start with a circle of radius a. Now place n points at random positions inside this circle. Can you calculate the expectation value for the mean distance between the points? For the sake of argument can you solve this for say n=5? This problem is very simple to solve computationally by just distributing points, measuring the distances and looping many times but I am looking for an analytical solution.