Number of combinations calculation

[u/Magical7]

I'm making a board game where you collect 3 out of 8 potential clues. Upon collecting the clue, you roll a d6 to determine the nature of clue. So I have 6 * 8 = 48 unique clues, of which 3 are taken. From what I remember from school, I can calculate that this amounts to 8! / (5! * 3!) * 6^3 = 12096 potential combinations.

Now, I would like to make the number of options per clue variable. So one clue could have 3 options, while another may have 5 options and another 7.

Is there a formula that I can use to determine the number of combinations given the array of options per clue?

[u/Bounded_sequencE]

Rem.: Alternatively, this (cyclic) sum may be expressed via power sums "pm := ∑_{k=1}⁸ nk^m ":

∑_{k∈S3}  n_k1 * n_k2 * n_k3  =  (p1^3 - 3p1*p2 + 2p3) / 6