$G(n,m)$ is the family of all graphs with $n$ vertices and $m$ edges (I consider $m < n$). Each graph in $G(n,m)$ is selected with uniform probability. What is the probability that the graph selected has exactly $c$ connected components?
An equivalent question is: what is the probability that exactly $k$ edges should be removed from the selected graph in order to make it a forest (graph without cycles)?
There is a solution for the case of $k = 0$ here.