P(A U B U C) = P(A) + P(B) + (C)+ - P(A intersect B) - P(B intersect C) - P (C intersect A) + P (A intersect B intersect C).
I don't know how to show this...
I assume you are familiar with the identity $\displaystyle \mathbb{P}(A \cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A \cap B)$.
Now
$\displaystyle \mathbb{P}((A \cup B) \cup C)=\mathbb{P}(A\cup B)+\mathbb{P}(C)-\mathbb{P}( (A \cup B) \cap C)$.
Expand this out again using the fact that $\displaystyle (A \cup B)\cap C=(A \cap C) \cup (B \cap C)$.