What does multiset imply?

Question

Does a multiset operation (($^n_k$)) only work for sets with identical elements, or does it also apply to replacement of distinct elements. Is there any difference in the calculation? Furthermore, is there any distinction between "choosing from a set" and the opposite action of "filling an order", where elements from an endless supply must be chosen to fit a set or sets with restrictions (quotes are for lack of a better phrase)? Are there different names for these two sides of the same concept? For reference, I think of the set chosen from as the "source", and the chosen set as the "selection". When we restrict the selection, it seems to use the same operations, but maybe not always. Can someone shed a little light on these ideas?

Answer 1

It would be better to start with the more familiar stars and bars formula, $\binom{n+k-1}{k-1}$ for putting $n$ identical objects into $k$ distinct bins

An example could be distributing $7$ identical ice cream bars to $5$ people, so (permitting $0$ assignments), the answer is $\dbinom{11}4$

But what if we want to buy a total of $7$ ice cream bars from a parlor having $5$ distinct varieties ? In other words, we can multichoose, buy any number of each variety subject to a total of $5$

This can be represented as $\left(\!\dbinom{k}{n}\!\right)$ to again give an answer of $\dbinom{11}{4}$

Notice that we have deliberately interchanged the meaning of $n$ and $k$ so as to get identical answers to both, which directly implies that putting $n$ identical items into $k$ distinct bins is the same as taking out a total of $n$ items from $k$ distinct types of objects.