Question 18
It is given that $\mathcal E=\{1,2,3,4,5,6,7\}$, and $A\subseteq\mathcal E$, $B\subseteq\mathcal E$, $C\subseteq\mathcal E$ are such that $A\cap C=\varnothing$ and $(A\cap B)\cup(B\cap C)=\{1,2,3,4\}$. Suppose that
$$S_1=A'\cap B'\cap C',\quad S_2=A\cap B',\quad S_3=B'\cap C,\quad S_4=A'\cap B\cap C'.$$
Find the number of possible ordered tuples of sets of the form $(A\cap B,B\cap C,S_1,S_2,S_3,S_4)$.
$$S_1=A'\cap B'\cap C',\quad S_2=A\cap B',\quad S_3=B'\cap C,\quad S_4=A'\cap B\cap C'.$$
Find the number of possible ordered tuples of sets of the form $(A\cap B,B\cap C,S_1,S_2,S_3,S_4)$.