5. (T-shirts, 10 points). A local charity sent an email to 20 volunteers saying that every volunteer who shows up on February 14 will get a free t-shirt. Each recipient of the email will show up with probability - (and decisions made by each volunteer do not affect other volunteers). The charity ordered 17 t-shirts. If more than 17 volunteers show up, then only 17 of them will get t-shirts, and the rest will not, because the charity will run out of them. Let random variable X be the number of volunteers that will show up on February 14. Express the following events in terms of X and calculate their probability (leave your answers as a neat expression rather than evaluating it). Out of the volunteers who show up on February 14, (a) exactly three will not get t-shirts, (b) exactly two will not get t-shirts, (c) at least one will not get a t-shirt. 2 6". (Optional, no collaboration, 10 points). BU announced a new policy that all students in CS 237 will have assigned seats for lectures. There are 100 seats in the classroom. All 100 students registered for the 2PM lecture are coming today. They will enter the classroom one at a time and take a seat. It turns out that the first student forgot which seat was assigned to him, so he will take a uniformly random seat. Each subsequent student entering the classroom will take their assigned seat if it is empty. However, if their seat is taken, they will take a uniformly random empty seat. What is the probability that the last student entering the classroom will end up in their assigned seat? (A complete solution to this problem should include a proof that the answer is correct.)