6. 10 balls are distributed among 3 urns. At each stage, a ball is selected at random from among all 10, taken from the urn it is in, and placed into one of the other two urns, each with equal probability. Let X,(n) be the number of balls in urn i, i = 1, 2,3 at time n, and consider the Markov chain X (n) = (Xi(n), X2(n), Xa(n)). (a) (5 points) What is the state space for this Markov chain? (b) (10 points) If the current state of the chain at time n is (a, b, c), what are the possible states of the chain at time n + 1, and what are the transition probabilities? (c) (10 points) Guess at the limiting probabilities for this Markov chain, and verify your guess by showing that the Markov chain is time reversible. (Note: Without loss of generality, you may consider just one pair of states (a1, by, ci) and (az, b2, (2) such that Plan,bei).(62,6z,ez) > 0.)