Factorial Notation

Show that $\frac{12!}{5!7!}+\frac{12!}{6!6!}=\frac{13!}{6!7!}$

Evaluate: $\frac{12!}{10!2!}$

If $S=1+1·\mathrm{1!}+2·\mathrm{2!}+3·\mathrm{3!}+...........n·n!$, then the value of $S$ is equal to

If $\frac{{\mathrm{ }}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{r}-1}}{\mathrm{a}}=\frac{{\mathrm{ }}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{r}}}{\mathrm{b}}=\frac{{\mathrm{ }}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{r}+1}}{\mathrm{c}}$ then

The sum of the factor of $8\mathrm{ }!$ which are odd and are of the form $\left(3\mathrm{m}+2\right)$ where $\mathrm{m}\in \mathrm{ }\mathrm{N}$ , is

The value of $2^n[1.3.5\dots \left(2n-3\right)\left(2n-1\right)]$ is