The Drunken Professor has a class of 34 students. He wants to divide them into seventeen pairs of partners for a project. He does so uniformly at random, with every possible division as likely as every other.
a) How many different ways are there to divide up the students?
(Example to avoid confusion: If there were six students, then the answer would be 15: AB/CD/EF,
AB/CE/DF, AB/CF/DE, AC/BD/EF, AC/BE/DF, AC/BF/DE, AD/BC/EF, AD/BE/CF,
AD/BF/CE, AE/BC/DF, AE/BD/CF, AE/BF/CD, AF/BC/DE, AF/BD/CE, AF/BE/CD.)
b) Two of the students are Alice and Betsy. What is the probability that they will be partners?
c) Suppose that the Drunken Professor has two projects (one before the midterm and one after). He divides up the class into pairs for each project, independently. Let X be the number of people who have the same partner for both projects. Compute the expected value of X.