Question

"if n balls are distributed randomly into k urns, what is the probability that the last urn contains j balls?"

Answer 1

The answer relies on whether the balls are different or not. If they are not, which is almost certainly what is intended.

If they are, the perceptive is a bit different. Your expression gives the likelihood that a particular set of j balls goes into the last urn and the other n−j balls into the other urns. But there are (nj) different possible sets of j balls, and each of them the same probability of being the last insides of the last urn, so the total probability of completing up with exactly j balls in the last urn is if the balls are different.

See attached file for the answer.