In many dice-games throwing 2 (3, 4...) of a kind or special combinations are meaningful events.

I am wondering how to approach the following problem:

Given d dice with s sides each. Of those s sides h sides count as a hit. The amount of hits shown after throwing the dice is n.

Searched is the probability for throwing a) n hits and b) n hits or (not exclusive) m hits with 1 < m < n.

Example with numbers: You got 10 dice with 6 sides each. How big is the chance to get exactly 4x "6" and how big to get 4, 3 or 2x "6".

It's related to this SE- post, but different.

I'm struggling to generlize the problem. Given 2 dice with 6 sides of whose 1 counts as a hit I got 1-(s-h/s)^d aka 1-(5/6)² and for given a bigger d I can use the Binomial coefficient to get all valid combinations (d! / (n! * (d-n)!)) but the whole clean formula is eluding me right now.

In case this makes any difference: This is no school-homework, I'm simply trying to balance a dice-game to play with some friends. I knew this back in school, but it's long gone now. I'm really grateful for any advice as it drives me crazy right now. :(


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