# Combinations with Repetition Versus … what is this?

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?

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
$$\binom{3+5-1}{5-1}=\binom74=35$$
• 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. – Suamere Dec 14 '15 at 17:57