$\left(\genfrac{}{}{0ex}{}{5}{2}\right)$ Proofs that $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\le \left(\genfrac{}{}{0ex}{}{n}{k+1}\right)$ if $k<n/2$

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.

arXiv commentTranscript of a mathematics colloquium talk delivered at Columbia University on Feb. 17, 2010