Prove that in a group of n>1 people there are two who have the same number of acquaintances in t

step 1 Ah, yes. So understanding the problem here, we can let the party have N people, where N is great

Prove that in a group of n>1 people there are two who have...

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Key Concepts

Pigeonhole Principle

This principle is a fundamental counting argument stating that if more objects (pigeons) are distributed among fewer containers (holes) than there are objects, then at least one container must hold more than one object. In the context of the problem, it is used to show that among the possible numbers of acquaintances, with more people than available unique acquaintance counts, at least two people must share the same number.

Graph Theory (Degree Sequence in Simple Graphs)

In graph theory, a simple graph models the situation by representing people as vertices and acquaintances as edges. The number of acquaintances a person has corresponds to the degree of the vertex. The conditions of the problem impose restrictions on the degree values, and the argument uses the fact that if one vertex has degree 0, then no vertex can have degree n?1. This relationship between the potential degrees and the structure of the graph is central to proving that a duplicate degree must exist.

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