I'll only provide a sketch and leave you to fill in the details.
First here's a handy counting formula that I always show my finite math students. For two sets $S$ and $T$,
$$ |S \setminus T | = |S| - |S \cap T| $$
where I write $S \setminus T$ instead of $S - T$ to represent the collection of objects that are in $S$ but not in $T$.
You should also know that if $S \cap T = S$, then $S \subseteq T$, and that $(S^C \cap T^C)^C = S \cup T$, by de Morgan's laws.
Now let's get to counting. Condition 2 says that
$$|U| - |C| = 150$$
Condition 3 says $$|A \cup B| = |A| + |B| - |A \cap B| = 90$$
$$|(A^C \cap B^C)^C| = |U| - |A \cup B| = 90$$
Finally condition 4 says
|(A \cup B) \setminus C| &= |A \cup B| - |(A \cup B) \cap C| \\
&= |A \cup B| - |(A \cap C) \cup (B \cap C)| \\
where I have used the distributive property of intersection over union. Now it's just up to some clever algebra.