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

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