How to extract a set $C$ that contains $N$ subsets of a set $B$, covers all elements of an external set $A$, but $N$ is minimal?
If you delete all subsets in $B$ that contain no string from $A$, then this is exactly the (Minimum) Set Cover problem, which is NP-hard, meaning that almost certainly no efficient (that is, polynomial-time) algorithm exists.

Testing each possible combination can be done in $O(m2^n)$ time if there are $n$ sets in $B$ and $m$ elements in $A$, which is better than factorial time.  (We don't care about the order of the subsets we select.)