Bipartite Graphs and Matchings · Challenge problem

Five committees need a representative, each from the following list of eligible people: committee $1$ from $\{1,2\}$, committee $2$ from $\{1,2,3\}$, committee $3$ from $\{2,3,4\}$, committee $4$ from $\{3,4,5\}$, committee $5$ from $\{4,5,1\}$ (numbering people $1$ through $5$). In how many ways can all five committees be given DIFFERENT representatives, each eligible for their own committee?