Say I have $s$ sets of colored balls, each set a different color, with $c$ balls in each set. I have $s$ bins, each with capacity $c$, in other words, when the balls are distributed completely, each bin must have precisely $c$ balls.
I want to calculate the probability that after distributing the balls uniformly randomly, respecting the capacity limit, each of the bins ends up full of one and only one color.
My thought is that this is the same as asking how many permutations of the totality of balls when partitioned into sets of size $c$ have those partitions all of distinct colors divided by the total number of permutations, so probability is $s!/((s c)!/c!^s)$
I'm not sure about that result, though: it seems like uniformly randomly assigning balls one at a time to bins not yet filled might be different than just counting the number of desired results over the total number of possible results, as in the probabilities for bin assignments change as bins fill.
How would one properly calculate this?