Coupon collection

There are $N$ distinct types of coupons in cereal boxes and each type, independent of prior selections, is equally likely to be in a box. If a child wants to collect a complete set of coupons with at least one of each type, how many coupons (boxes) on average are needed to make such a complete set?
Now suppose the child has collected $n$ coupons. What is the expected number of distinct coupon types among them? Give this expected number for $N=3$ and $n=3$.