2965288

$$|\bigcap_i A_i| = \sum_{i} |A_i| - \sum_{i<j}|A_i\bigcup A_j| + \sum_{i<j<k }| A_i\bigcup A_j\bigcup A_k| - \dots$$

every element that belongs to all $$A_1...A_n$$ may be found exactly once in the left intersection. In the right-hand part it is counted multiple times, like :

$$n\ times - \binom n 2 \ times + \binom n 3 \ times =\cdots = 1 \  time $$ because $$(1-1)^n = 0$$.

Overall, every element in intersection is counted exactly one time so we get the size of the intersection.