# Finding cardinality of sets without Venn diagrams

So I'm studying Probability and I'm having a hard time figuring out how to find the cardinality of unions and intersections given certain information. Specifically, given a sample space $\Omega$ and three events $A,B,C$ :

\begin{gathered} n(\Omega ) = 270 \hfill \\ n(A) = n(B) = n(C) = 90 \hfill \\ n(A \cap B) = n(A \cap C) = n(B \cap C) = 30 \hfill \\ n(A \cap B \cap C) = 10 \hfill \\ \end{gathered}

How do I find the cardinality of the intersection of the three complements? $n({A^c} \cap {B^c} \cap {C^c}) \\$

I could solve it very easily drawing a Venn diagram but I want to know how to solve it analytically.

$$n(A^c\cap B^c\cap C^c) = n((A\cup B\cup C)^c)=n(\Omega)-n(A\cup B\cup C)$$
Use PIE to calculate $n(A\cup B\cup C)$