This is a homework question I was assigned to do. I was thinking about the problem and I thought the answer was $ \frac{1}{k^j} \cdot \left(1 - \frac{1}{k}\right)^{n-j} . $ I now have some doubts, though. I was thinking that I am not taking into account that the balls could be distributed into the urns in different combinations, so perhaps I have to include some expression involving ${{A}\choose{B}}$, where $A$ and $B$ are some expressions involving $n, k$ or $j$ ?
Thanks in advance.