I know it's a combinatorics question, I'm just not sure how to approach it.
Let's say we have $7$ types of agents: $A,B,C,D,E,F,G$. Agents work in teams of $4$. We send teams on missions, with each mission requiring three different types of agents. For example, $\{ A,B,C \}$. So, the team of $ \{ A,B,C,D \}$ can do four types of missions: $\{ A,B,C\}$, $\{ A,B,D \}$, $\{ A,C,D \}$ and $\{ B,C,D \}$.
The question is: how many teams of $4$ do I need to cover all possible missions? And what are these teams?
My progress so far: I know the upper bound is 35 (just create a team for every mission as "mission requirements + any other member"), and the lower bound is 35 \ 4 ~= 9 (each team covering almost exactly 4 different missions, and I have an intuition it's impossible to construct). As a programmer, my first way of approaching this would be "just simulate every combination of teams", and I might do just that if other approaches fail. I'm interested in how to do it mathematically. My combinatorics classes were a long time ago.
Here's a somewhat intuitive 12-team solution:
A computer search reveals that there's no 11-team solution, so this is optimal.