2019 AMC 8 Problems/Problem 25
Alice has apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?
We use stars and bars. Let Alice get apples, let Becky get apples, let Chris get apples. We can manipulate this into an equation which can be solved using stars and bars.
All of them get at least apples, so we can subtract from , from , and from . Let , let , let . We can allow either of them to equal to , hence this can be solved by stars and bars.
By Stars and Bars, our answer is just .
Without loss of generality, let's assume that Alice has apples. There are ways to split the rest of the apples with Becky and Chris. If Alice has apples, there are ways to split the rest of the apples with Becky and Chris. If Alice has apples, there are ways to split the rest. So the total number of ways to split apples between the three friends is equal to
Let's assume that the three of them have apples. Since each of them has to have at least apples, we say that and . Thus, , and so by stars and bars, the number of solutions for this is - aops5234
We can give each person one apple first so that apples are shared between the three people, where each person receives at least one apple. Using Stars and Bars, the number of ways to do this is .
Since we are giving each person at least apples anyway, lets put aside. Now we have more apples to distribute to everyone. Since the apples are all indistinguishable, we can put "dividers" in between the apples. Ex. There are ways. -SigmaPiE
Videos explaining solution
https://www.youtube.com/watch?v=aStuVhoD8wc- Also includes other problems from 21-25
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