2008 AMC 10A Problem 23

Two subsets of the set S=\lbrace a,b,c,d,e\rbrace are to be chosen so that their union is S and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?

\mathrm{(A)}\ 20\qquad\mathrm{(B)}\ 40\qquad\mathrm{(C)}\ 60\qquad\mathrm{(D)}\ 160\qquad\mathrm{(E)}\ 320

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.