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Say I have $n$ objects and 100 small buckets. I assign the objects to buckets at random.

Each small bucket can fit at most 5 objects in. To accommodate a 6th, the small bucket must be switched out with a medium bucket that can fit in 10 objects. To accommodate an 11th, this again must switched out for a large bucket that can fit 15 objects. This switching out is repeated with increasingly large buckets that can hold an increasing number of objects, multiple of 5.

What is the probability density function of the number of switches required to accommodate $n$ objects?

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