Haven't been able to find something similar, hope this isn't redundant.
I've managed to widdle down a problem to a k balls in n boxes (boxes and balls being distinguishable). All assumptions are pretty classic (equiprobability, without exclusion, etc.) and I am looking to find the probability of no boxes containing exactly one ball.
I thought I'd arrive to a meaningful result by using Jordon's formula and being a bit smart about defining the events, but I've been hitting dead ends for days. I was wondering if a combinatorial approach could work. It is not very far from trivial when looking for how many combinations leave no box empty, I imagine it has to be possible. I worked a bit on trying to count those scenarios with very little success. If anyone could confirm or deny that the approach is possible or suggest a new one I'd be happy to dig it :)
Thanks in advance, Cheers!
//EDIT
To make the problem a bit more interesting, how about we divide the k balls into two categories (let's go for j black ones and k-j white ones) and look for the probability that no black ball is alone in one box.
To clarify, two black balls in one box do not count, as 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.
Any ideas/novel approaches from here?
Thanks again!
