An instructor gives an exam with sixteen questions. Students are allowed to choose any eleven to answer. (Hint: To answer the following questions, these examples may be helpful: Examples 9.5.4, 9.5.5, 9.5.6, and 9.5.7.) (a) How many different choices of eleven questions are there? 4368 (b) Suppose seven questions require proof and nine do not. (0) How many groups of eleven questions contain five that require proof and six that do not? 1764 (ii) How many groups of eleven questions contain at least one that requires proof? 4368 (iii) How many groups of eleven questions contain at most three that require proof? 175 x (c) Suppose the exam instructions specify that at most one of questions 1 and 2 may be included among the eleven. How many different choices of eleven questions are there? 4368 (d) Suppose the exam instructions specify that either both questions 1 and 2 are to be included among the eleven or neither is to be included. How many different choices of eleven questions are there?
