Find an average number by using group action In a party there are $n$ individuals. Each of them gives a present in the party. At the end of the party the presents are distributed randomly to each participant. How many individuals receive in average their own presents?
We want to solve the problem from the perspective of a particular group action $f: G \times M \rightarrow M$ and a fixed set, $M^g = \{ x \in M\mid g\cdot x = x \}$. I do not know how to go from here.
Do you have any suggestion or a solution proposal? Thanks.

All this boils down to is a shuffling of the presents, and the question how many fixed points.
We have $S_n$ acting on $n$ points.
By Burnside's lemma, $$\lvert M/G\rvert =1/\lvert G\rvert \sum \vert M^g\rvert. $$  The number of orbits is the average  number  of  fixed points.
We get $$\lvert M/G\rvert =1,$$ because,  as we know, the action is transitive.
