_________________ _ _ (A & B) /\ (C \/ B) \/ (C & B)

To simplify the logical expression \((A \land B) \land (C \lor B) \lor (C \land B)\), we can break it down step by step using logical identities and rules.

1. Expression to Simplify: \[ (A \land B) \land (C \lor B) \lor (C \land B) \]

2. Apply the Distributive Law: Let's simplify \((A \land B) \land (C \lor B)\): \[ (A \land B \land C) \lor (A \land B \land B) = (A \land B \land C) \lor (A \land B) \] (Since \(B \land B\) is just \(B\)).

3. Combine the Results: Now substituting this back into the original expression: \[ (A \land B \land C) \lor (A \land B) \lor (C \land B) \]

4. Factor Out the Common Terms: Notice \(B\) is common in both \((A \land B)\) and \((C \land B)\): \[ B \land (A \lor C) \lor (A \land B) \]

5. Combine Like Terms: The expression can be simplified further: \[ (A \land B) \lor (C \land B) = B \land (A \lor C) \]

Thus, the final simplified expression is: \[ B \land (A \lor C) \]

In summary, the logical expression \((A \land B) \land (C \lor B) \lor (C \land B)\) simplifies to: \[ B \land (A \lor C) \]