I'm trying to derive a result for the number of possible combinations of r objects from n, when we have unlimited numbers of objects to select from. (Wikipedia suggests that these are called multi-combinations). My current line of reasoning is like this:
Suppose we can choose 4 letters from an unlimited set containing {A, B, C, D, E}. Then we can partition this into choices where 4 are identical, 3 are identical, 2 are identical and all are different, like so:
combs like AAAA + combs like AAAB + combs like AABB + combs like AABC + combs like ABCD
Now, it seems to me that to go this route, I'll have to start enumerating the partitions of r (here 4) to ensure that I'll get all possible combinations with r, r-1, r-2, ... objects. This looks tricky to me.
Can someone tell me a) if this is not a fruitful approach and b) suggest a hint (and no more than a hint please) as to a more straightforward method, as at the moment I don't see one ?