Question: The Head of the Department of Mathematical Sciences at a certain university has 12 mathematicians, seven computer scientists, and three statisticians in his employ. He wishes to appoint some committees from among these 22 people.
(a) How many five-member committees can he appoint?
(b) How many five-member committees, each containing at least one statistician, can he appoint?
(c) A certain professor of mathematics, Dr. G, and a certain colleague, Dr. P, refuse to serve together on the same committee. How many five-member committees can be formed so as not to contain both Dr. G and Dr. P?
(d) How many five-member committees can be formed so that the number of mathematicians is greater than the number of computer scientists and the number of computer scientists is greater than the number of statisticians?