Let ${\mathrm{P}}_{\mathrm{n}}$ denote product of binomial coefficients in ${\left(1+\mathrm{x}\right)}^{\mathrm{n}}$ then $\frac{{\mathrm{P}}_{\mathrm{n}+1}}{{\mathrm{P}}_{\mathrm{n}}}$

$\begin{array}{rcl}{\mathrm{P}}_{\mathrm{n}}& =& {}^{\mathrm{n}}{\mathrm{c}}_0.{}^{\mathrm{n}}{\mathrm{c}}_1.{}^{\mathrm{n}}{\mathrm{c}}_2....{}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{n}}\end{array}$

$\begin{array}{rcl}\frac{{\mathrm{P}}_{\mathrm{n}+1}}{{\mathrm{P}}_{\mathrm{n}}}& =& \frac{{}^{\mathrm{n}+1}{\mathrm{c}}_0.{}^{\mathrm{n}+1}{\mathrm{c}}_1....{}^{(\mathrm{n}+1)}\mathrm{c}_{\mathrm{n}+1}}{{}^{\mathrm{n}}{\mathrm{c}}_0.{}^{\mathrm{n}}{\mathrm{c}}_1.{}^{\mathrm{n}}{\mathrm{c}}_2....{}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{n}}}\end{array}$

$\begin{array}{rcl} & & \\ & & \\ \frac{{\mathrm{P}}_{\mathrm{n}+1}}{{\mathrm{P}}_{\mathrm{n}}}& {=}^{\mathrm{n}+1}{\mathrm{C}}_{0 }\left(\frac{{}^{(\mathrm{n}+1)}\mathrm{c}_1}{{}^{\mathrm{n}}{\mathrm{c}}_0}\right) \left(\frac{{}^{(\mathrm{n}+1)}\mathrm{c}_2}{{}^{\mathrm{n}}{\mathrm{c}}_1}\right).....\left(\frac{{}^{(\mathrm{n}+1)}\mathrm{c}_{\mathrm{n}}}{{}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{n}}}\right)& \\ & & \end{array}$

Since $\boxed{{}^{\mathrm{n}+1}{\mathrm{c}}_{\mathrm{r}+1}=\frac{\mathrm{n}+1}{\mathrm{r}+1}.{}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{r}}}$

$\frac{{\mathrm{P}}_{\mathrm{n}+1}}{{\mathrm{P}}_{\mathrm{n}}}=\frac{{(\mathrm{n}+1)}^{\mathrm{n}}}{\mathrm{n}!}$