Say I have a set of $n$ distinct objects and I want to divide it into $k$ identical boxes each of which will has exactly $r_i$ objects, $1\leq i \leq k$. How many ways can I do it?
I guess that the idea is this - one can choose the "first" $r_1$ of these in $^nC_{r_1}$ ways and the next $r_2$ of them in $^{(n-r_1)}C_{r_2}$ ways and so on till $^{n-\sum_{j=1}^{j=k-1}r_j}C_{r_k}$. But if $a_i$ of these $r_i$s are of size $i$ then those $a_i$ can be shuffled among themselves in $a_i!$ ways. Hence the final answer is,
$\frac{^nC_{r_1}\prod_{j=1}^{j=k-1}^{n-\sum_{m=1}^{m=j} r_m}C_{r_{j+1}} }{\prod _{i=1}^n a_i !} = \frac{n!}{(\prod_{j=1}^k r_j !)(\prod_{i=1}^n a_i!)}$
Is this right? If it is right then is there a ``higher" way to get this answer - which might be more insightful?
Is it possible to think of the above as counting the number of solutions of some (Diophantine?)equation?
If this is right is there a natural relationship of this answer to Bell's number or Faa di Bruno's formula?
