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I'm back adding more layers to my combinatoric problem to serve as instruction (Black and white balls, probability that no black ball is alone).

The scenario is still the same and straightforward: k distinguishable balls ($j$ black and $k-j$ white) go in $n$ distinguishable boxes with an equal probability without exclusion.

I would this time find the probability of $m$ boxes containing exactly one lone black ball. Using the answers I found for the same problem where all the balls are the same color and help I got for my simpler scenarios, I started by counting the number of ways to arrange the balls so that the $l$th box (we numbered them beforehand) contains one lone black ball:

$$N(l)=j(n−1)^{(k-1)}$$ By extension $$N(l1,l2,..,lo)=j(j-1)...(j-o+1)(n-o)^{(k-o)}$$ with $l1$, $l2$,..., $lo$ being distinct

I do not really understand how to apply inclusion-exclusion for this formula to give the metric I am looking for. I would gladly take any explanation to it :)

If you also know other methods or approaches that arrive to the solution (more or less directly), I'd be happy to try and work on them!

Thanks in advance!

//Emitted formula formatting

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    $\begingroup$ Please use MathJax to format your post. $\endgroup$
    – saulspatz
    Jun 10, 2019 at 14:05
  • $\begingroup$ Hope it suits now $\endgroup$
    – AlexM
    Jun 10, 2019 at 14:27
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    $\begingroup$ I don't understand the question. You say "$m$ boxes contain one lone black ball." Does this mean that at least $m$ boxes contain exactly one back ball each, or that exactly $m$ boxes contain exactly on black ball each, or that $m$ boxes chosen at random each contain exactly one black ball each, or is it perhaps that the $m$ boxes contain one black ball in total? Please make the question more explicit. Also, you seem to be saying that there's another problem somewhere, that you are incorporating by reference. If so, please post a link to that question. $\endgroup$
    – saulspatz
    Jun 10, 2019 at 14:53
  • $\begingroup$ The metric I would like to end up with is the probability of exactly $m$ boxes each containing exactly one lone black ball (meaning 1 black ball and 0 white balls). $\endgroup$
    – AlexM
    Jun 10, 2019 at 17:48

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