Subcontests
(4)\sum_{(E_1,...,E_k) \in E_k }|E_1 \cap ... \cap E_k|
Let X be a set of n elements and k be a positive integer.
Consider the family Sk of all k-tuples (E1,...,Ek) with Ei⊆X for each i.
Evaluate the sums ∑(E1,...,Ek)∈Sk∣E1∩...∩Ek∣ and ∑(E1,...,Ek)∈Sk∣E1∪...∪Ek∣