${5\choose 2}$ Proofs that ${n\choose k} \leq {n\choose {k+1}}$ if $k<n/2$

TL;DR

This paper presents ten distinct proofs of the combinatorial inequality that choosing k out of n elements is less than or equal to choosing k+1 out of n elements when k is less than half of n, highlighting the richness of mathematical reasoning.

Contribution

It introduces ten different proofs of a fundamental combinatorial inequality, showcasing diverse mathematical techniques and perspectives.

Findings

Multiple proofs of the inequality ${nrace k} \, \leq \, {n\brace k+1}$ for $k < n/2$

Demonstrates the depth and variety of approaches in combinatorial mathematics

Highlights the non-trivial nature of seemingly simple mathematical facts.

Abstract

There is no trivial mathematics, there are only trivial mathematicians! A mathematician is trivial if he or she believes that there exists trivial mathematics. Being a non-trivial mathematician myself, I will describe ten different proofs of the seemingly trivial fact that the number of ways of choosing k people out of n people is less than or equal to the number of ways of choosing k+1 people out of n people, provided that k is less than half of n.