Given that I am surrounded by Mathematicians here, let me ask for help for what should be a simple problem I can't seem to be able to solve:
Assume you have n fair dice with m faces (i.e. each can roll an integer from 1 to m with a uniform probability). You roll all n, and keep the k (with 0<k<=n) highest results. What is the probability that the sum of the k dice you kept is X?
(If one keeps all the dice, probability-generating functions give the answer straightforwardly. If I roll 2 dice and keep 1 I can easily enumerate the outcomes and calculate the probabilities, but I am stumped by the general case).
@j_bertolotti so I think this is an equivalent formulation: instead of taking the top k dice out of n, calculate the probability of the value X from the sum of k dice with m sides (use the probability-generating function like you said) and then multiply the probability of the remaining n-k dice having all values less than or equal to the smallest value of the k dice.
Let me know if this formulation is correct or not and if it is helpful.
@sojournTime I need one more coffee in me before I can properly think about it, but it feels similar to how I first approached the problem (getting completely nonsensical results). Will let you know 🙂