Binomial Coefficient expressed using Beta Function

Theorem

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

Then:

$\dbinom r k = \dfrac 1 {\paren {r + 1} \map B {k + 1, r - k + 1} }$

$\ds \dbinom r k$

$=$

$\ds \dfrac {r!} {k! \, \paren {r - k}!}$

$\ds $

$=$

$\ds \dfrac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} }$

$\ds $

$=$

$\ds \dfrac {\map \Gamma {r + 2} } {r + 1} \dfrac 1 {\map \Gamma {k + 1} \map \Gamma {r - k + 1} }$

$\ds $

$=$

$\ds \dfrac 1 {r + 1} \dfrac {\map \Gamma {r + 2} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} }$

rearranging

$\ds $

$=$

$\ds \dfrac 1 {r + 1} \dfrac 1 {\map B {k + 1, r - k + 1} }$

Definition 3 of Beta Function

$\blacksquare$