Binomial Coefficient expressed using Beta Function

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\dbinom r k$ denote a binomial coefficient.

Then:

$\dbinom r k = \dfrac 1 {\left({r + 1}\right) B \left({k + 1, r - k + 1}\right)}$


Proof

\(\ds \dbinom r k\) \(=\) \(\ds \dfrac {r!} {k! \, \left({r - k}\right)!}\) Definition 1 of Binomial Coefficient
\(\ds \) \(=\) \(\ds \dfrac {\Gamma \left({r + 1}\right)} {\Gamma \left({k + 1}\right) \Gamma \left({r - k + 1}\right)}\) Gamma Function Extends Factorial
\(\ds \) \(=\) \(\ds \dfrac {\Gamma \left({r + 2}\right)} {r + 1} \dfrac 1 {\Gamma \left({k + 1}\right) \Gamma \left({r - k + 1}\right)}\) Gamma Difference Equation
\(\ds \) \(=\) \(\ds \dfrac 1 {r + 1} \dfrac {\Gamma \left({r + 2}\right)} {\Gamma \left({k + 1}\right) \Gamma \left({r - k + 1}\right)}\) rearranging
\(\ds \) \(=\) \(\ds \dfrac 1 {r + 1} \dfrac 1 {B \left({k + 1, r - k + 1}\right)}\) Definition 3 of Beta Function

$\blacksquare$

Sources