We are given three events \(A\), \(B\), and \(C\) with certain properties and conditions. Let's break down the information and find the value of \(P(C)\).

1. The events \(A\) and \(B\) are disjoint, meaning \(P(A \cap B) = 0\).
2. The events \(A\) and \(C\) are independent, implying \(P(A \cap C) = P(A) \cdot P(C)\).
3. The events \(B\) and \(C\) are independent, meaning \(P(B \cap C) = P(B) \cdot P(C)\).
4. We also know that \(4P(A) = 2P(B) = P(C)\).
5. Finally, \(P(A \cup B \cup C) = 5P(A)\).

Let's denote \(P(A) = x\). Therefore:

• \(P(A) = x\)
• \(P(B) = 2x\)
• \(P(C) = 4x\)

Using the formula for the union of three events, we have:

\(P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)\)

Since \(A\) and \(B\) are disjoint, \(P(A \cap B) = 0\). Also, \(P(A \cap B \cap C) = 0\) because \(A\) and \(B\) are disjoint.

Substituting values, the equation becomes:

\(5x = x + 2x + 4x - (2x \cdot 4x) - (x \cdot 4x)\)

We simplify this to:

\(5x = 7x - 8x^2 - 4x^2\)

\(5x = 7x - 12x^2\)

Rearranging the equation gives:

\(12x^2 - 2x = 0\)

Factoring out \(x\), we get:

\(x(12x - 2) = 0\)

Since \(x \neq 0\) (as probabilities are non-zero for non-null events), we have:

\(12x - 2 = 0 \implies 12x = 2 \implies x = \frac{2}{12} = \frac{1}{6}\)

Thus, \(P(C) = 4x = 4 \times \frac{1}{6} = \frac{2}{3}\).

Hence, the value of \(P(C)\) is \(\frac{2}{3}\).