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Question 1 Suppose we throw k balls into m bins, by choosing for each ball a bin uniformly and independently at random. The balls are labeled with distinct numbers born 1 to k and the bins with distinct numbers from 1 to m. (a) Let S g {1, . . . , is}. What is the probability that all balls in S are thrown into the same bin? Express this probability in terms of |S| and m. (13) Give an upper bound for the probability that there exists a bin that contains three or more balls. Your upper bound should use the union bound, and be as small as possible. Explain and show all your calculations