Problem 5: 16 students are randomly assigned to four different dorm rooms, with four students in each.

Answer the following:

(a) Suppose eight of the students are clean, and eight of the students are messy. Use a counting argument to find the probability that no dorm room will contain both a messy and a clean student. (You do not have to use a counting argument for any future part.)

(b) What is the expected number of rooms that will contain both a messy and a clean student?

(c) Let X be the largest number of clean students in any room, and Y be the largest number of messy students in any room. Are X and Y independent? Why or why not? (Do not use a qualitative argument, use the definition of independence to definitively show that they are independent or not.)