Count ways to place $n$ identical balls into $n$ urns so that exactly one urn is empty? 
How many ways are there to plane n indistinguishable balls into n urns so that exactly one urn is empty?

Why is the answer for this question n(n-1)?

It does not matter which urn is empty, and there are $n$ urns, so we have $n$ choices of which urn to leave empty.
Next, every other urn must have at least one ball in it (since we can only have one empty urn), so must distribute $n-1$ of the balls into these $n-1$ urns.  We have one ball left and we can choose any of those $n-1$ urns to put the last ball into (it cannot go into the empty urn, or it would no longer be empty).
Thus, we have $n$ choices of which urn is empty, and then $n-1$ choices of where to place the extra ball, giving $n \cdot (n-1)$ total ways to accomplish this.
