upper and lower bounds to binomial coefficient

Given two integers $n,k>0$ such that $k\le n$, we have the following inequalities for the binomial coefficient $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$:

$\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)$ $\le$ ${\displaystyle \frac{n^k}{k!}}$

$\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)$ $\le$ ${\left({\displaystyle \frac{n\cdot e}{k}}\right)}^k$

$\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)$ $\ge$ ${\left({\displaystyle \frac{n}{k}}\right)}^k$

Here $e$ is the base of natural logarithms. Also, for large $n$, $\left(\genfrac{}{}{0pt}{}{n}{k}\right)\approx \frac{n^k}{k!}$.

Title	upper and lower bounds to binomial coefficient
Canonical name	UpperAndLowerBoundsToBinomialCoefficient
Date of creation	2013-03-22 13:29:53
Last modified on	2013-03-22 13:29:53
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	6
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 05A10