No there is misunderstanding about what I tried to say.You have 3 parameters: number of urns (u), number of balls (b) and maximum balls in any urn (m). (I think it's been established that the size of the urns does not matter).
For any specific u, b and m, you can solve it using my technique, but it can get very long and tedious.
It was okay for u = 14, b = 20 and m = 2. But when m = 3, it would take a lot of calculations.
If you are looking for a general formula invovling u, b and m, I think that would be hard.
Any complicated formula assuming that the formula exists could be simplified to give NOT the exact result but a result nearing the exact solution.
How to afford a problem like this one by simplifying the solution.
We have 14 urns we want to place 20 balls on those urns and we wnat to know what is the probability that one urn contains al least 3 balls (or 4 balls).
How can we restate the problem without altering or changing it and find out the solution quickly using a simple reasoning?