Suppose a lot of n objects contains $n_1$  , objects of one kind, $n_2$   objects of second kind $n_3$   objects of third kind,..... $n_k$ objects of k^th kind, Such that $n_1+n_2+\mathrm{....}{n}_k=n,$  then the number of possible arrangements/permutations of r objects out of this lot is the coefficient of  $x^r$  in the expansion of  $r!\prod _{ }\left[\sum _{\lambda =0}^{n_i}\frac{x^{\lambda }}{\lambda !}\right]$  on the basis of above information, answer the following