Imagine that you're working at a large chain bookstore, and you've just
gotten a hundred copies of the latest bestseller. You are assigned to
shelve them. No problem, except that each shelf only holds at most
thirty books. Again, no problem: you can put 30 each on three
shelves, and 10 on a fourth shelf. Or you could put 25 each on four
shelves... or maybe 20 each on five shelves... or if you were feeling
really disgruntled, you could put one copy on each of a hundred
different shelves, scattered around the store.
Now, in case you weren't already disgruntled, the manager tells you that
if none of the books sell on a given day, you are to reshelve them
differently the next day, and again the next day, until they start
selling. How many ways are there to arrange these things?
(Hint: try small numbers first!)