Connections in a state

A certain fictional state has 12 cities in it, each with railroads emanating from them. The cities have (in no particular order) $1,1,2,2,3,3,4,4,4,5,8,9$ Railroads emanating from them. Define a Connection to be a path from $A$ to $B$ through a certain city $C$ . $A$ and $B$ do not have to be distinct (i.e. sightseers going to a city then returning), though $A \rightarrow C \rightarrow B$ is distinct from $B \rightarrow C \rightarrow A$ . Find the number of possible connections in this state.