Mastering the Combinations Problem: Buying a Dozen Donuts from 5 Types

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The discussion focuses on calculating the number of ways to buy a dozen donuts from five types, emphasizing that the order of selection does not matter. Initially, participants explore combinations for one and two donuts, noting the increasing complexity as more donuts are added. A key point is the distinction between combinations and permutations, as the order of donut types does not create unique combinations. The correct approach involves using the "stars and bars" theorem, leading to the conclusion that the number of ways to choose 12 donuts from 5 types is given by the formula for combinations with repetition. The final consensus highlights the importance of understanding the difference between unique arrangements and combinations in this problem.
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Can someone explain to me how to do this problem... I am really lost...

how many ways can you buy a dozen donuts from an unlimited supply of 5 types of donuts?
 
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Ok there are 12 donuts you want , and 5 possible types of donuts.

so if you were to get only one donut, there would only be 5 possible combinations.

if you were to get 2 donuts, there would be 25 possible combinations...

11 12 13 14 15, 21 22 23 24 25, 31 32 33 34 35, 41 42 43 44 45, 51 52 53 54 55.

Now the problem there is some numbers are repeated. i.e 21=12, but according to your problem 21 and 12 are different ways (it does not say unique ways).

so 1 donut = 5, 2 donuts = 25. starting to see the patern?
 
so the answer would be 5^12?
 
Yes, that is correct.
 
No, I don't think that's correct. Buying doughnut type 4 and doughnut type 5 is not different from buying type 5 and type 4.
 

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I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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