Having a little bit of trouble with this question, but I don't necessarily want the answer, I'm looking for an explanation on how to do it, and if my theory is correct.
How many ways are there to place 25 different books on 10 number shelves in the following scenarios:
a) Order of books on a shelf matters b) Order of books on a shelf doesn't matter c) Order matters, but each shelf must have at least one book
Firstly, I want to make sure I'm understanding the question correctly. We have 10 shelves, and for a, I want to see how many orderings of books there are, where like one shelf could have all 25 potentially?
Here's my thinking:
a) $25P25$ for the first place, as order matters and there's 25 ways the books could be arranged on that shelf, or it could be in the second place, third, etc. so $(25P25)*10$?
Or would I do it like a Stars and Bars question? Because the above answer is massive.
b) Order doesn't matter, so combinations now. Would I do $25C25 * 10$? Or is there a way to differentiate Stars and Bars between permutations and combinations?
c) Stars and Bars but with the restriction of each gap having one at least?