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Question: After another gym class you are tasked with putting the 27 identical dodgeballs away into 4 bins. This time, no bin can hold more than 7 balls. How many ways can you clean up? So I believe this is a problem of over counting, as I currently have

C(30,3) - [C(4,3) C(11,3) - C(4,4) C(3,3)]

But this is not that answer. Any help is appreciated, as I need to better understand this material.

Kenyon
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    "no bin can hold more than 7 balls" happens to be very, very constricting. I would focus on that. – Arthur Mar 03 '21 at 21:53

2 Answers2

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If no bin can contain more than $7$ balls, and there are only $4$ bins, that means at most $28$ balls can be placed in the bins, and that this maximum is uniquely attained when each bin contains the maximum of $7$ balls. Since we have only one less than this maximum number, it means that the only ways to place $27$ balls in the bins is to find the number of ways to remove one ball from the maximum configuration. And since there are only $4$ ways to do this--in each case, picking the bin from which to remove a single ball--the answer is $4$.

heropup
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  • Thank you. Just to make sure I understand this, if there were to be 5 bins instead of 4, the answer would change to 5. If there were to be 4 bins but 26 balls, would the answer change to C(5,2)? – Kenyon Mar 03 '21 at 22:08
  • @RandomUser55 If there are 5 bins and 34 balls, sure, the answer is $5$. If there are 5 bins and 27 balls, you start getting a lot of alternatives. – Arthur Mar 03 '21 at 22:12
  • If there are five bins, but the maximum is still $7$ balls per bin, then the question becomes much more complicated, because now there are many more possible ways to distribute $27$ balls; e.g., $(5,3,7,6,6)$, or $(1,7,7,7,5)$. It would be $5$ ways if you had to distribute $7(5)-1 = 34$ balls. – heropup Mar 03 '21 at 22:13
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Hint: if there were 28 balls instead of 27, each by would need to have $\frac{28}{4}=7$ balls in it, so there would only be one possible arrangement. Therefore, this problem is equivalent to asking how many ways you can remove 1 ball from one of the bins.

Sandejo
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