Question

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

Answer 1

1 56₂ 7C7 + 5 C37C6 > 80 c) 1,2 included 12-2=10 9-2 -7 10c7 = 120 or d) either 1,2 included neithee 1,2 included. LOC++ loca > 130