Question 30
For any set $S,$ let $|S|$ denote the number of elements in $S,$ and let $n(S)$ be the number of subsets of $S,$ including the empty set and the set $S$ itself. If $A,$ $B,$ and $C$ are sets for which $$
\begin{gathered} n(A)+n(B)+n(C)\\ {}=n(A\cup B\cup C),\\ |A|=|B|=100, \end{gathered}
$$ then what is the minimum possible value of $|A\cap B\cap C|?$
\begin{gathered} n(A)+n(B)+n(C)\\ {}=n(A\cup B\cup C),\\ |A|=|B|=100, \end{gathered}
$$ then what is the minimum possible value of $|A\cap B\cap C|?$