My discrete mathematics book gives the following definition for the pigeonhole principle:(adsbygoogle = window.adsbygoogle || []).push({});

If m objects are distributed into k containers where m > k, then one container must have more than [itex]\lfloor[/itex][itex]\frac{m-1}{k}[/itex][itex]\rfloor[/itex] objects.

It then states as a corollary that the arithmetic mean of a set of numbers must be in between the smallest and largest numbers of the set. No proof is given, it pretty much just says "well it's just obvious that this is the case."

I think it is obvious that the arithmetic mean of a set of numbers is in between its smallest and largest values. What isn't obvious to me is how their definition of the pigeonhole principle leads to the corollary. Can anyone help me out?

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# Pigeonhole Principle and the Arithmetic Mean

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