The Quiz Problem) Consider a quiz contest where a person is given a list of N questions and can answer these questions in any order he chooses. Question i will be answered correctly with probability pi, independently of earlier answers, and the person will then receive a reward Ri. At the first incorrect answer, the quiz terminates and the person is allowed to keep his previous rewards. The problem is to maximize the expected reward by choosing optimally the ordering of the questions.

(a) Show that to maximize the expected reward, questions should be answered in decreasing order of piRi/(1 - pi

(b) Consider the variant of the problem where there is a maximum number of questions that can be answered, which is smaller than the number of questions that are available. Show that it is not necessarily optimal to answer the questions in order of decreasing

(c) Give a 2-OPT algorithm to solve the problem where the number of available questions is one more than the maximum number of questions that can be answered.