To determine the correct Boolean expression for the given circuit, let's analyze the circuit step by step:

1. The circuit consists of three input variables: \(x\), \(y\), and \(z\).
2. The circuit has several logic gates, including NOT, AND, and OR gates.
3. First, let's identify the outputs of each gate:
   • AND Gate 1: Inputs are \(\bar{x}\), \(\bar{y}\), and \(z\). The output is \(\bar{x}\bar{y}z\).
   • AND Gate 2: Inputs are \(\bar{x}\), \(y\), and \(z\). The output is \(\bar{x}yz\).
   • AND Gate 3: Inputs are \(\bar{y}\) and \(z\). The output is \(\bar{y}z\).
4. The outputs of these AND gates are then combined using an OR gate. Therefore, the final output \(F\) is:
   • \(F = \bar{x}\bar{y}z + \bar{x}yz + \bar{y}z\)

This matches the given correct answer option. The correct Boolean expression for the circuit is:

\(F = \bar{x}\bar{y}z + \bar{x}yz + \bar{y}z\)