Verified answer

Do you have this theorem?

P(AUB)=P(A) + P(B) - P(A∩B)

Start with that. But instead of B, use the set BUC.

P(AU(BUC))=P(A) + P(BUC) - P(A∩(BUC))

Now use the theorem again to write the expression for P(BUC).

What do you do with P(A∩(BUC))? Well A∩(BUC) = (A∩B) U (A∩C), yet another set union. So use the theorem one more time with those two sets (A∩B) and (A∩C).

It can be visualized if you use vend diagrams (overlapping circles)