Upper Bound for Binomial Coefficient

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

Then:
 * $\dbinom n k \le \left({\dfrac {n e} k}\right)^k$

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

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


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

so that:


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

Then:

Hence the result.