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In how many ways can $4$ balls be placed in $8$ boxes if neither the balls nor the boxes are distinguishable?

 Oct 4, 2023
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When the balls are indistinguishable, there is only one way to place them in the boxes. We can simply put one ball in each box.

However, the boxes are also indistinguishable, so there are 8! ways to arrange the boxes.

Therefore, there are 8!​ ways to place 4 balls in 8 boxes if neither the balls nor the boxes are distinguishable.

 Oct 4, 2023

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