# 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 $ | $\frac{{n}^{k}}{k!}$ | ||

$\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)$ | $\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 |