Ok, here is what I am attempting to find an answer to: 


I draw M uniformly random subsets of size K from the set of numbers $\Omega=\{1, \dots, N\}$ (where uniformly random means that each unique subset of $\Omega$ is equally likely). What is the probability that $S \leq M$ subsets share $R \leq K$ elements? 
I am especially interested in the case when $M>2$ and $K>2$, or really just a general formula that incorporates every parameter.



Probability of k overlapping subsets in N trials
To be precise: given $S \in \{2,\ldots,M\}$ and $R \in \{0,\ldots,K\}$ you want the probability $P(S,R)$ that the intersection of some $S$ of your $M$ random sets has cardinality $R$. 


Well, I'll do the case $M=3$, $S=2$. Let your random sets be $X\_1, X\_2, X\_3$.
Of course you need $N + R \ge 2 K$.


$$ P(|X\_1 \cap X\_2| = R) = \dfrac{{N \choose R, K-R, K-R,N+R-2K} }{{N \choose K}^2} = \dfrac{K!^2\; (N-K)!^2}{N! \;R! \;(K-R)!^2\; (N-2K+R)!}$$


$$ \eqalign{P(|X\_1 &\cap X\_2| = |X\_1 \cap X\_3| = R)\cr = &\sum\_{j=\max(0,2R-k)}^R \sum\_{i=\max(0,3K-N-2R)}^{\min(R,K-R)} \dfrac{{N \choose i, j, R-j, R-j,K-2R+j,K-R-i,K-R-i,N-3K+2R+i}
}{{N \choose K}^3} \cr
}$$


$$\eqalign{P(|X\_1 &\cap X\_2| = |X\_1 \cap X\_3| = |X\_2 \cap X\_3| = R) \cr
= & \sum\_{j=\max(0,2R-K)}^{\min(R,N-3K+R)} \dfrac{{N \choose j,R-j,R-j,R-j,K-2R+j,K-2R+j,K-2R+j,N-3K+R-j}}{{N\choose K}^3}}$$


And then $$ P(2,R) = 3 P(|X\_1 \cap X\_2|=R) - 3 P(|X\_1 \cap X\_2| = |X\_1 \cap |X3|=R) + P(|X\_1 \cap X\_2| = |X\_1 \cap X\_3| = |X\_2 \cap X\_3| = R) $$


For $M > 3$ and $S > 2$, things get even more complicated.


