Union of $3$ Sets Formula Derivation

Question

How to derive the formula for a union of 3 sets by using the rules for a union of 2 sets?

What I have so far:

\begin{align} P(A\cup B\cup C) &= P((A\cup B)\cup C) \text{ by commutative property}\\ &= P(A\cup B) + P(C) - P((A\cup B)C) \\ &= P(A) + P(B) - P(AB) + P(C) - P((A\cup B)C)\\ & = \ldots ? \\ &= P(A) + P(B) + P(C) - P(AB) - P(AC) - P(BC) + P(ABC) \end{align}

Answer 1

Notice that \begin{align}-P((A \cup B)C)&=-P(AC \cup BC)\\ &=-[P(AC)+P(BC)-P(AC \cap BC)] \\ &=-P(AC)-P(BC)+P(ABC)\end{align}

Answer 2

\begin{align} P(A∪B∪C) &= P((A \cup B) \cup C) \\ &= P(A \cup B) + P(C) - P((A \cup B)C) \\ &= P(A) + P(B) - P(AB) + P(C) - P((A \cup B)C) \\ &= P(A) + P(B) - P(AB) + P(C) - P(AC \cup BC) \\ &= P(A) + P(B) - P(AB) + P(C) - P(AC) - P(BC) + P(ACBC) \\ &= P(A) + P(B) + P(C) - P(AB) - P(AC) - P(BC) + P(ABC) \\ \end{align}