$$\begin{array}{rcl}\text{f}\left(k;n,p\right)& =& {}_n{C}_k{p}^k{\left(1-p\right)}^{n-k}\\ & =& \frac{n!}{k!\left(n-k\right)!}{p}^k{\left(1-p\right)}^{n-k}\\ & =& \frac{\Gamma \left(n+1\right)}{\Gamma \left(k+1\right)\Gamma \left(n-k+1\right)}{p}^k{\left(1-p\right)}^{n-k}\\ & =& \frac{p^k{\left(1-p\right)}^{n-k}}{B\left(k+\mathrm{1,}n-k+1\right)\left(n+1\right)}\end{array}$$