Find the n best choices while maximizing value Given: 


A list of slots (A, B, C, ...)
Every slots supports a list of choices (C0, C1, C2, C3...)
Every choice has a value 


All slots must be filled with at most n different choices. The sum of the values should be maximized.

Example:

Choice-Values: 


C0 = 4
C1 = 3
C2 = 2
C3 = 1


Slots: 


A: C1, C3
B: C0, C1, C3
C: C2, C3
D: C0, C1, C2, C3
E: C1, C2, C3


Solution for n = 1: C3, C3, C3, C3, C3

Solution for n = 2: C1, C1, C2, C1, C1

Question 

Is this a known problem? 

It reminds me of a mix between the knapsack problem and of a scheduling problem, as I want to get the maximum value while filling a certain space using a limited set of items.
Thanks to Yuvals comment i found this site discussing the set cover problem. It's exactly what I needed.