In a large University with k academic departments, we must appoint an important committee. One professor will be chosen from each department. Some professors have joint appointments in two or more departments, but each must be the designated professor representative of at most one department. We must use equally many assistant professors, associate professors, and full professors among the chosen representatives (assume that k is divisible by 3). How can the committee be found? (Hint: Build a network in which units of flow correspond to professors chosen for the committee and capacities enforce tthe various constraints. Explain how to use the network to test whether such a committee exists and find it if it does.)