What you are asking about are called covering designs. A more standard notation in combinatorics defines $(v,k,t)$-covering design to mean:
$v =$ total number of points (your universe N)
$k =$ size of subsets (blocksize, your set size X)
$t =$ size of covered combinations (your K elements)
There is a special case where each t-subset is covered exactly once, and these are known as Steiner systems. Obviously in those cases the number of blocks used is a minimum.
The minimum number of blocks ($k$-subsets) needed to cover all $t$-subsets of a "universe" of size $v$ is termed $C(v,k,t)$, and there is no general formula for it. In fact not much is known for values of $t$ more than ten.
There is a well-known general lower bound for $C(v,k,t)$ called the Schönheim inequality:
$$C(v,k,t) \ge \lceil (v/k) * C(v-1,k-1,t-1) \rceil$$
which can be applied recursively down to the case $t=1$, where trivially:
$$C(v,k,1) = \lceil v/k \rceil$$
I think it is known that this lower bound is fairly tight for sufficiently large $v$. The La Jolla Covering Repository (linked above) has some good papers on constructing covering designs, but they seems to have gotten hidden when the site underwent a redesign awhile back. I'll see if I can ferret them out and post those lost links.
Added: Lost Links
[New Constructions for Covering Designs -- Gordon et al, 1995]
[Asymptotically Optimal Covering Designs -- Gordon et al, 1995]
[Handbook of Combinatorial Design, chapter excerpt by Gordon and Stinson]