We have n distinguishable balls (say they have different labels or colours). If these balls are dropped at random in n boxes, what is the probability that:
1- No box is empty? 2- Exactly one box is empty?
For 1, I figured that we have $n^n$ ways to put the n balls into the n boxes. And I figured there are $n!$ to sort the balls so there is one ball for each box.
So is the answer to question 1 $n!/(n^n)$?
For 2, there are $n-1$ ways for the boxes to be empty. This is because you can have box 1 be empty (and just that), box 2 be empty and just that, so ultimately you can have at most $n-1$ variations of empty boxes.
So I figured the solution to part 2 was $\frac{n-1}{n^n}$
Is any of this right?