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Problem. How many ways are there to put $5$ balls in $3$ boxes if the balls are not distinguishable but the boxes are?

How many ways are there to put $n$ balls in $k$ boxes if the balls are not distinguishable but the boxes are? Generalize.

I don't really get how to deal with these problems, involving Distinguishability. Any solutions?

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  • $\begingroup$ "Stars and bars" $\endgroup$ – JMoravitz Aug 30 '20 at 2:10
  • $\begingroup$ The quick, handwavy explanation... arrange five stars and two bars in a row. The bars act as a sort of barrier. Things to the left of the leftmost bar go in the first box. Things to the right of the rightmost bar go in the last box. Things between the bars go in the middle box. We only used two bars instead of three because they represent changing from one box to the next which only happens twice. $\endgroup$ – JMoravitz Aug 30 '20 at 2:12

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