Suppose a lot of n objects contains ${\mathrm{n}}_1$ objects of one kind, ${\mathrm{n}}_2$ objects of second kind, ${\mathrm{n}}_3$ objects of third kind,....., ${\mathrm{n}}_{\mathrm{k}}$ objects of kth kind. Such that ${\mathrm{n}}_1+{\mathrm{n}}_2+{\mathrm{n}}_3+\dots {\mathrm{n}}_{\mathrm{k}}=\mathrm{n},$ then the number of possible arrangements/permutations of r objects out of this lot is the coefficient of ${\mathrm{x}}^{\mathrm{r}}$ in the expansion of $\mathrm{r}!\prod \left(\sum _{\mathrm{\lambda }=0}^{{\mathrm{n}}_1} \frac{{\mathrm{x}}^{\mathrm{\lambda }}}{\mathrm{\lambda }!}\right)$ on the basis of above infromation, answer the following questions: