Re: Can this be solved in polynomial time?

You have a set S of size N.
You are given M subsets of S.
Find K of the given subsets, so that their union is of maximal size.

Can this problem be solved efficiently? If so, how?

This is not homework.

Ok. Thanks.

I do not think it can be solved efficiently. Proof by reduction to the
set covering decision problem. WLOG we can assume that the union of
the M subsets is all of S. If you could solve your optimization problem
quickly, you could also answer the following question quickly:

Does there exist K of the given subsets, whose union is all of S?

(Just run your algorithm for your optimization problem, and see whether
the K subsets it produces cover all of S.) But the latter question
is exactly the set covering decision problem, which is known to be
NP-complete.
.