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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:

bbb,bbc,bbl,bbs,bbv,bcb,bcc,bcl,bcs,bcv,blb,blc,bll,bls,blv,bsb,bsc,bsl,bss,bsv,bvb,bvc,bvl,bvs,bvv,cbb,cbc,cbl,cbs,cbv,ccb,ccc,ccl,ccs,ccv,clb,clc,cll,cls,clv,csb,csc,csl,css,csv,cvb,cvc,cvl,cvs,cvv,lbb,lbc,lbl,lbs,lbv,lcb,lcc,lcl,lcs,lcv,llb,llc,lll,lls,llv,lsb,lsc,lsl,lss,lsv,lvb,lvc,lvl,lvs,lvv,sbb,sbc,sbl,sbs,sbv,scb,scc,scl,scs,scv,slb,slc,sll,sls,slv,ssb,ssc,ssl,sss,ssv,svb,svc,svl,svs,svv,vbb,vbc,vbl,vbs,vbv,vcb,vcc,vcl,vcs,vcv,vlb,vlc,vll,vls,vlv,vsb,vsc,vsl,vss,vsv,vvb,vvc,vvl,vvs,vvv

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?

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If you’re building an ice cream cone, banana-chocolate-lemon is different from banana-lemon-chocolate, and there are indeed $5^3=125$ possibilities, because the order makes a noticeable difference. If you’re just dumping three scoops of ice cream into a dish, there is no meaningful difference between banana-chocolate-lemon and banana-lemon-chocolate, and in that case there are only

$$\binom{3+5-1}{5-1}=\binom74=35$$

possibilities. If neither situation is clearly specified, the problem is ambiguous, though my inclination would be to assume that the order does not matter (i.e., that we’re in the dish situation) unless a cone or similar structure is explicitly mentioned.

My best advice is not to worry about specific terms like combination with repetition, but rather to analyze what is actually going on in any given problem.

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  • $\begingroup$ I love your analysis that it depends if you're in a dish or cone. Point still, those are two different situations with two different equations. One equation is named "x" and the other equation is named "y"? I was teaching daughter about combinations allowing repetition. She took initiative and came to different answers online. So I'd like to explain to her the difference between Stars/Bars Versus the super-basic n^k Is there no term to differentiate them? I can't even find an article online that says n^k is even an option because all "combinations allowing repetition" lead to stars/bars. $\endgroup$ – Suamere Dec 14 '15 at 17:57
  • $\begingroup$ @Suamere: I think that I’d emphasize that what makes the difference here is whether we care about the order. In both cases here we care about the number of scoops of each flavor, but in only one of them to we care about the order. You could also say that when we care about the order, two scoops of the same flavor are no longer indistinguishable: first scoop lemon, second scoop lemon, and third scoop lemon are three different ‘kinds’ of lemon scoop. $\endgroup$ – Brian M. Scott Dec 14 '15 at 18:00
  • $\begingroup$ Yes. As in Base10, if there were 0-9 flavors, 999 would mean that one 9 is worth 100, one is worth 10, and one is worth 1. There are a total of 1000 posibilities of 3-scoop cone. Because position matters. When googling around with your additional perspective, I found that "combination" is incorrect, the descriminating term is "Permutations" when the order matters. $\endgroup$ – Suamere Dec 14 '15 at 18:14
  • $\begingroup$ @Suamere: Yes, permutation implies that order matters in some way. $\endgroup$ – Brian M. Scott Dec 14 '15 at 18:15

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