5.5 This problem has arisen in evaluating random backgrounds in an experiment involving many counting channels. Suppose one has three channels and each of the three channels is excited randomly and independently. Suppose at any instant the probability of channel i being on is pi. Imagine “on” means a light on some instrument panel is on. To simplify the problem, take p1 = p2 = p3 = p. At random intervals, pictures are taken of the instrument panel with the lights on it. In a given picture, what are the probabilities of 0, 1, 2, 3 channels being on? N total pictures are taken. What is the joint probability that n0 pictures have 0 channels on, n1 have 1 channel on, n2 have 2 channels on, and n3 have 3 channels on, where we must have n0 + n1 + n2 + n3 = N?