${5\choose 2}$ Proofs that ${n\choose k} \leq {n\choose {k+1}}$ 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.