Here is a "colourful" description of what I would like to count. Suppose you have one of those tables you see in a casino. I think they are for roulette, with $m$ squares, each of them with a number inside them. You have $c_1$chips of type $c_2$ chips of type $2\dots$ $c_n$ chips of type $n$. You have to place the chips on top of the squares such that each number has at least $1$ chip over it but no square has two or more chips of the same colour on top of it. In how many ways can we do this?