Harvard Statistics 110: see #30, p. 30 of pdf
Randomly, k distinguishable balls are placed into n distinguishable boxes, with all possibilities equally likely. Find the expected number of empty boxes.
I'm not convinced by the solution because, as illustrated by the picture of balls and boxes, there are $n^k$ different ways of filling the boxes by the balls, and there are $k$ different ways of one box being filled. One box can be filled by $1, 2, \dots k$ balls. Then, $P(I_j=0)=k/n^k$, and $P(I_j=1)=1-k/n^k$. But this will give weird results for $n=9, k=13$. I don't know but I feel there's something missing. May someone help?