I've brought a extra layer to a problem I posted here: K balls in N boxes, no boxes contain 1 ball
Nota bene: This is a problem I'm using to work on my combinatorics
The scenario is pretty classic: k distinguishable balls (j black and k-j white) go in n distinguishable boxes with an equal probability without exclusion.
The objective is to find the probability that no black ball is left alone in a box. To clarify, two black balls in one box do not count, as does not a white ball alone in a box. The only scenario that counts is a black ball being alone in any box.
Here is the approach I've been using: We start by placing the black balls at random in the n boxes (non exclusive) and then count possible arrangements of white balls not leaving any black ball alone using occupancy vectors. Only problem is that occupancy do not account for the variety of the balls, or at least how I've been using them. Would it be a good idea to try and adapt those vectors and use a combination of multiple ones or rather find a more direct approach maybe using multinomial coefficients? (As I said earlier, combinatorics is really not my cup of tea but I'd like to learn the toolbox to solve most of the 'basic' problems).
Thanks in advance, Cheers!