how can I calculate the probability to get triples or better when throwing n 6-sided dice? I've been banging my head on a wall with this question. I'm designing a game and would like to implement a loot system inspired by a game called "Vermintide" where players roll a certain number of dice and gain loot according to the result.
I want my players to get rewards depending on the result of the dice. but I need to evaluate the quality of items, and the number of dice they get to throw based on the probability.
Essentially if they get doubles they get a standard item, if they get triples they get a magic item etc...
I heard of the "birthday problem" and "multinomials" but most of what I could find about it seemed to be very particular cases (I found birthday problem for 2 or more but not for 3 or more, which seems much less trivial).
Is there a smart way to go about this problem which I first thought was going to be trivial, and the more I search the more it seems complicated.

If I understand your problem correctly, you're looking for a generalization of the binomial distribution to $n > 2$ where $n$ is the number of possible classes. The term for this is a multinomial distribution.
We can view a $n = 2$ side analog of a dice as a coin. The probability of a single outcome is a Bernoulli distribution (e.g. $\Pr(H)$). The probability of multiple outcomes is a Binomial distribution (e.g. $\Pr(HT)$).
For $n > 2$, the probability of a single outcome is a Categorical distribution (e.g. $\Pr(1)$). The probability of multiple outcomes is a Multinomial distribution (e.g. $\Pr(1126)$).
Remember that a multinomial distribution will only tell you the distribution of, say 5 of the rolls being $1$. So if you're looking for pairs or triplets regardless of the class -- whether its 1 or 2, you don't care -- you have to multiply your probability (if its a fair dice) or construct a sum in the case the dice is not fair.
