Consider a random experiment of distributing r indistinguishable balls to n different cells (for simplicity, let us index the cells as 1, 2. . .. , n). Suppose that all distinguishable arrangements are equally likely to occur (For example, in the case of r = 2 and n = 2, then the three arrangements (2, 0), (1, 1), and (0, 2) have equal probabilities). This setting is closely related to the Bose-Einstein statistics in quantum physics. (a) Let X be the number of balls in the Ist cell (and X is clearly a random variable), and let q, be the probability of the event that X = k. Show that for any k ( NU{0), 9k = Contr-k-2/contr-1 Please clearly explain each step of your answer. (b) Based on the result in (a), show that if the average number of particles per cell r tends to A as n - co and r -+ co, then we have q - X"/(1+ ))*+ as n -> co and r -> co. What kind of random variable is X in the limit