A multiple-choice quiz consists of 5 independent questions. Each question has 4 options with exactly one correct answer. A student, who has not studied, guesses the answer to every question at random. Let C be the event "the student answers at least one question correctly". Work out the following:
a. (i) Determine the probability that the student answers none of the questions correctly.
b. (ii) Using the result from part (i), find P(C).
c. (iii) If the quiz were to contain 8 such independent questions instead of 5, how would the probability in part (ii) change? Calculate the new probability that the student answers at least one question correctly.