There are n students applying to n colleges. Each college has a ranking over all students (i. e. a permutation) which, for all we know, is completely random and independent of other colleges. College number i will admit the first ki students in its ranking. If a student is not admitted to any college, he or she might file a complaint against the board of colleges, and colleges want to avoid that as much as possible. (a) If for all i, ki = 1 (i. e. if every college only admits the top student on its list), what is the probability that all students will be admitted to at least one college?
(b) What is the probability that a particular student, Alice, does not get admitted to any college? Prove that if the average of all ki’s is at least 2lnn, then this probability is at most 1/n . (Hint: use the inequality 1−x ≤ e−x)
(c) Prove that when the average ki is at least 2lnn, then the probability that at least one student does not get admitted to any college is at most 1/n.