An instructor gives an exam with sixteen questions. students are allowed to choose any eleven to answer.

(a) how many different choices of eleven questions are there? 1365 correct: your answer is correct.
(b) suppose six questions require proof and nine do not.
(i) how many groups of eleven questions contain four that require proof and seven that do not? 540 correct: your answer is correct.
(ii) how many groups of eleven questions contain at least one that requires proof? 1365 correct: your answer is correct. (iii) how many groups of eleven questions contain at most three that require proof? 176 incorrect: your answer is incorrect.
(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?
(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?