A number of different objects have been distributed into n...

A number of different objects have been distributed into $n$ boxes $B_{1}, B_{2}, \ldots, B_{n} .$ All the objects from these boxes are removed and redistributed into $n+1$ new boxes $B_{1}^{*}, B_{2}^{*}, \ldots, B_{n+1}^{*}$, with no new box empty (so the total number of objects must be at least $n+1$. Prove that there are two objects each of which has the property that it is in a new box that contains fewer objects than the old box that contained it.

Key Concepts

Proof by Contradiction

Proof by contradiction is a logical technique where one assumes the negation of the statement to be proven and then shows that this assumption leads to a logical inconsistency or violation of known facts. In combinatorial distribution problems, assuming that fewer than the required number of objects have moved to boxes with lower counts can lead to impossible totals or distributions upon summing over all boxes, thereby forcing the original claim to be true.

Averaging Arguments

Averaging arguments rely on the idea that the total sum of objects divided by the number of groups gives an average occupancy. When individual box counts deviate significantly from this average, it guides us to conclude that some boxes must have less than, and some more than, the average. This discrepancy is used to deduce that at least a couple of objects must end up in boxes with a lower count than what they started with.

Pigeonhole Principle

The pigeonhole principle is a foundational concept in combinatorics stating that if objects are distributed among a number of containers, then at least one container must hold at least a certain minimal number of objects whenever the number of objects exceeds the number of containers. In redistribution problems, this principle often helps to guarantee that some boxes or groups, after redistribution, will necessarily contain fewer objects than expected based on the original distribution, leading to useful inequalities.

Double Counting

Double counting is a method where one counts the same quantity in two different ways to derive an equality or inequality. In such problems, one might sum the numbers of objects over the original groups and again over the redistributed groups, revealing relationships between the counts in the association and ultimately showing that some elements must fall into boxes with different occupancy profiles.

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