Assume that we have $p$ sets, with given sizes: $m_1,m_2,...,m_p$. The (distinct) elements in each set are taken from $N$ elements (where $m_1,m_2,...,m_p \le N$).A combination is defined as an assignment of distinct elements (from $N$ possible ones) to each of the sets. For example, say that we have $p=2$ sets of sizes $m_1=2,m_2=3$ and that $N=4$. Then one possible combination is $\left\{ {1,2} \right\},\left\{ {1,2,3} \right\}$, another one is $\left\{ {2,3} \right\},\left\{ {2,3,4} \right\}$ and so on. It is easy to note that there are $\prod\limits_{i = 1}^p {\left( {\begin{array}{*{20}{c}}N\\{{m_i}}\end{array}} \right)} $ possible combinations.My question is - given the sizes $m_1,m_2,...,m_p$, how many combinations have intersection of exact size $k=1,2,...,{\text {min}}(m_i)$ (denote this by $I(k)$)?My thought was to find first how many combinations have an intersection of size at least $k=1,2,...,{\text {min}}(m_i)$ (denote this by $v(k)$), and then $I(k) = v(k) - v(k+1)$.So far I know that for $k_{min}={\text {min}}(m_i)$:$$v\left( {{k_{\min }}} \right) = \left( {\begin{array}{*{20}{c}}  N \\   {{k_{\min }}} \end{array}} \right)\prod\limits_{i = 1}^p {\left( {\begin{array}{*{20}{c}}  {N - {k_{\min }}} \\   {{m_i} - {k_{\min }}} \end{array}} \right)} $$(and $I(k_{min})=v(k_{min})$).However, the formula above for $k<k_{min}$ gives an over estimate. Any help would be greatly appreciated.
Intersection between sets
co.combinatorics;np