Proof of ${n \choose 0},{n \choose 1},{n \choose 2},...,{n \choose \lfloor n/2\rfloor}$ is monotonically increasing
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.