# upper and lower bounds to binomial coefficient

Given two integers $n,k>0$ such that $k\leq n$, we have the following inequalities for the binomial coefficient ${n\choose k}$:

 $\displaystyle{n\choose k}$ $\displaystyle\leq$ $\displaystyle\frac{n^{k}}{k!}$ $\displaystyle{n\choose k}$ $\displaystyle\leq$ $\displaystyle\left(\frac{n\cdot e}{k}\right)^{k}$ $\displaystyle{n\choose k}$ $\displaystyle\geq$ $\displaystyle\left(\frac{n}{k}\right)^{k}$

Here $e$ is the base of natural logarithms. Also, for large $n$, ${n\choose k}\approx\frac{n^{k}}{k!}$.

Title upper and lower bounds to binomial coefficient UpperAndLowerBoundsToBinomialCoefficient 2013-03-22 13:29:53 2013-03-22 13:29:53 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Theorem msc 05A10