Maybe I'm calling my problem something it isn't. When I search for "Combinations with Repetition" I get answers like this: Wikipedia
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It seems as if there are physically five exactly like balls, and two boxes. So the combinations could be:
5,0 | 4,1 | 3,2 | 2,3 | 1,4 | 0,5
So supposedly that is called Combinations with Repetition? It is explained everwhere as such. Then what is it called if you say you have three different types of balls (a,b,and c), and they can be combined in any number of ways in two different containers?
a,a | a,b | a,c | b,a | b,b | b,c | c,a | c,b | c,c
That, to me, is a list of all combinations, allowing repetition. But nowhere online does it say to take k types into n containers, and do n^k
One citation that I find specifically incorrect is this:
Given that we have five different flavors of icecream we could choose from, and we are going to get THREE SCOOPS... how many combinations can we have?
banana (b), chocolate (c), lemon (l), strawberry (s) and vanilla (v)
The way I see it, I could have bbb, vvv, or bvs. These are the combinations:
That is, 125 possible combinations, or n^k, 5^3
However, the answer on this site says there are only 35 combinations of ice cream because it uses the stars and bars theorems, which I believe is completely wrong for the situation.
So how do you differentiate between the two situations? Are they not both called combinations with repetition? Or are there seperate terms to describe each?