Question 6
Let $A$ be a set with $|A|=225$, meaning that $A$ has $225$ elements.
Suppose further that there are eleven subsets $A_1, \dots, A_{11}$ of $A$
such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$
for $1\leq i\lt j\leq11$.
Prove that $|A_1\cup A_2\cup\dots\cup A_{11}|\geq 165$,
and give an example for which equality holds.
Suppose further that there are eleven subsets $A_1, \dots, A_{11}$ of $A$
such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$
for $1\leq i\lt j\leq11$.
Prove that $|A_1\cup A_2\cup\dots\cup A_{11}|\geq 165$,
and give an example for which equality holds.