I have a set $S$ of $N$ elements; then I do three independent samplings from $S$, and I create three sets $R_1$, $R_2$ and $R_3$ of sizes $N_1$, $N_2$ and $N_3$. Let $N_{int}$ be the size of the intersection of the three, i.e., the cardinality of the set of elements of $S$ that are also in $R_1$, $R_2$ and $R_3$.
I would like to associate to $N_{int}$ a p-value, for example computing the probability of having a greater intersection if the three subset were choosen randomly.
If I had only two of such sets (say $R_1$ and $R_2$) I would have used the hypergeometric distribution; how can I extend to this case?
I have been trying by bootstrapping, but since my observation is very significant it would require to many iterations to beat it.