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An instructor gives an exam with seventeen questions. Students are allowed to choose any twelve to answer. (a) How many different choices of twelve questions are there? (b) Suppose seven questions require proof and ten do not. (i) How many groups of twelve questions contain five that require proof and seven that do not? (ii) How many groups of twelve questions contain at least one that requires proof? (iii) How many groups of twelve questions contain at most three that require proof? (c) Suppose the exam instructions specify that at most one of questions 1 and 2 may be included among the twelve. How many different choices of twelve questions are there? (d) Suppose the exam instructions specify that either both questions 1 and 2 are to be included among the twelve or neither is to be included. How many different choices of twelve questions are there