🚀 go-pugleaf

Who can help with the following problem...

SOGGIE, breakfast of chumps, is having a special promotion. They are packing
a special 'prize' in each box of SOGGIES. There are three different prizes in
the collection, distributed uniformly among the boxes, and each box contains
one prize.
You, constant reader, buy one box at a time.

     a. What is the probability of getting all three prizes in exactly three
boxes? That is, after buying three boxes, each box contains a different
prize.

     b. What is the probability of getting all three prizes in exactly four
boxes?

     c. What is the probability of getting all three prizes in exactly K
boxes?

     d. What is the average (expected) number of boxes needed to get all
three prizes?

     e. If the number of prizes in the collection is N, find a function E(N),
the expected number of boxes required to get all N prizes in the collection.

I've assigned this problem situation to my AP computer science class to
program as a simulation using Pascal 'sets' and they can graph a good
approximation of the function through Monte Carlo methods, but I've not yet
come up with a function E(N). I have determined that
(a) P(3 boxes) for three prizes is 2/9 (that was easy: 1*2/3*1/3)
and
(b) P(4 boxes) for three prizes is 2/7 (by inspecting the outcome space, but
I am stuck even for a formula for (b).

Any insights are always appreciated!
JohnHanna@aol.com