Finding the Top Coders You are putting together a programming team and trying to recruit the best of the best for your team. To do so, you organize a head-to-head Coding Tournament. Each programmer will participate in a head-to-head coding battle with each of the other programmers, so that every person battles against every other competitor. Thus, for every pair of programmers, the tournament will determine a winner between the two. At the end of the tournament, you will select 3 programmers for your programming team which will be named the typedef terrapins. It will be rather embarrassing for you if there is some programmer who beat all 3 of your typedef terrapins in the tournament, but who was not selected for the team. We will say that your team is dominant if it avoids this situation. (a) If n programmers compete, how many different head-to-head coding battles will there be? Your answer may depend on the number of teams n.
(b) If n programmers compete, how many possible teams could you have? Your answer may depend on the total number of teams n.
(C) Suppose that there are n programmers in the tournament, and they are equally skilled so that the winner of any given head-to-head battle is a random event: each of the two programmers in a head-to-head battle will win with probability 1/2. After the tournament, you select three of the programmers, which you are considering declaring the winners. A programmer named Jane is not one of these three. Given that the outcome of each contest was random, what is the probability that Jane beat all 3 of the programmers you chose? What's the probability that Jane didn't beat all 3 programmers?
(d) You again have a set of 3 programmers that you consider selecting for your team, the typedef terrapins. If the outcomes of the head-to-head contests are random as in the previous part, what is the probability that this group of 3 programmers would make a dominant team? That is, what is the probability that none of the n-3 other programmers beat all 3 of these chosen programmers?