A problem concerning counting

throneoo

I understand the ambiguity here . What I mean in that example is that there are two categories of books , there are infinite books for each category . What matters is the no. of categories of books , but not the quantity for each category. For instance , there are 2 kinds of books : Category A and Category B . Each category of books are enough to fill up infinite boxes . Then there will be 5 different ways to fill up a box that can contain 4 books . Which are :

AAAA
AAAB
AABB
ABBB
BBBB

throneoo

wow..thanks a lot for providing such satisfying answers

haruspex

Yes, 3060. But there's a much easier way.
Think of a box as containing four locations for books, and between them, and at the ends of the box, slots for very thin dividers, 5 slots in all. There are 14 dividers to go in each box and any number of them can occupy the same slot. When packing the box, left to right, a divider tells you to switch to the next book type. So the placement of dividers tells you exactly how a box is to be packed, and conversely, for any given choice of books for a box, arranging them in a canonical order determines the appropriate way to place the dividers; so this is a one-to-one mapping.
Now consider the book locations and dividers as just 18 things in a row. Four are book locations, 14 are dividers. Number of possible arrangements is 18 choose 4 = 3060.