Step 1: Understanding the Concept:
The {lac} operon is a regulated gene system in \(E.\ coli\). The \(i\) gene (inhibitor gene) produces a repressor protein that normally binds to the operator to block transcription. When lactose (the inducer) is present, it binds to the repressor, changing its shape so it can no longer bind to the operator, allowing transcription of the structural genes (\(z, y, a\)).
Detailed Explanation:
In this specific mutation, the \(i\) gene product (repressor) is altered such that it can no longer bind to the inducer molecule (lactose/allolactose).
1. Because the repressor cannot bind to the inducer, the presence of lactose in the medium does not affect the repressor's state.
2. The repressor remains permanently bound to the operator region of the operon.
3. This block prevents RNA polymerase from transcribing the structural genes \(z, y, a\).
4. Consequently, no mRNA is produced, and the enzymes needed to metabolize lactose are not synthesized (not translated).
Step 2: Final Answer:
Since the repressor is stuck on the operator, the \(z, y, a\) genes will not be expressed. Option (C) is the correct outcome.