Lower Bound for Binomial Coefficient

Theorem
Let $n, k \in \Z$ such that $n \ge k \ge 0$.

Then:
 * $\dbinom n k \ge \paren {\dfrac {\paren {n - k} e} k}^k \dfrac 1 {e k}$

where $\dbinom n k$ denotes a binomial coefficient.

Proof
From Lower and Upper Bound of Factorial, we have that:


 * $k! \le \dfrac {k^{k + 1} } {e^{k - 1} }$

so that:


 * $(1): \quad \dfrac 1 {k!} \ge \dfrac {e^{k - 1} } {k^{k + 1} }$

Then:

Hence the result.