Lots of number theoretic inequalities are to be had from the binomial theorem. I remember reading the below argument as part of Erdos's proof of Bertrand's postulate:

Suppose that $n$ is a positive integer, then we have

$$4^n = (1 + 1)^{2n} = {\sum_{j=0}^{2n}}{2n\choose{j}}.$$

Thus, since $ 2n\choose{n}$ is the maximum value of the sequence $({2n\choose{k}})$, we conclude that

$$ 4^n < (2n + 1){2n\choose{n}}.$$

I thought it was neat.