(f) how many ways are there to place a total of m indistinguishable balls into n distinguishable urns, with some urns possibly empty or with several balls?

The number of different ways to distribute m indistinguishable balls into n distinguishable urns is given by C(m+n-1,n-1).

For example, the number of ways 6 ball can be placed in 7 boxes is given by C(6+7-1, 7-1) = C(12, 6) = 924 ways.