# Stars and bars: estimating maximum distance between bars

I have, say, 24 objects spread across 10 boxes. Each object can independently fall into any box (equal probability to any box).

Stars and bars let's me estimate number of possible combinations. But what's the probability distribution of number of objects in the fullest box?

Or to be more concrete, what's the possibility that at least one box will contain at least 6 objects? Or exactly one with exactly 6 objects?

I suspect that I could replace !******! with an apple to estimate the probability of at least one box with exactly 6 objects but what about the former questions? Are these solvable by stars'n bars?

• What is the probability space? Is an object equally likely to be in any box? Are the positions of different objects independent? Or do you mean that the different distributions (as computed by stars and bars) are equally likely? – saulspatz Nov 24 '18 at 14:49
• @saulspatz objects are equally likely to be in any box. I edited the question. – Džuris Nov 24 '18 at 14:58
• Nice problem. I don't think stars and bars will help, though. Suppose we have ten objects and ten bins. Stars and bars gives equal weight to the outcome with all ten objects in the first bin, and to the outcome with all objects in one bin, which doesn't help for your problem at all. – saulspatz Nov 24 '18 at 15:15