Which of the following Logix5000 controller modes will allow the modification of codes in a running plant?

The option indicating Remote Run Mode is the correct choice because it allows for modifications to be made to the program while the process is running. In this mode, you can remotely access and make changes to the program of the Logix5000 controller without interrupting its operational status. This capability is essential for making adjustments or updates without incurring downtime.

In contrast, other modes such as Run Mode do not allow changes to the program while the controller is actively executing code. This ensures system stability and safety during normal operations, as real-time data processing and execution should not be disrupted by code alterations. Similarly, Remote Program Mode is for uploading or modifying the program while the controller is not running, and Remote Test Mode is meant for testing changes in a non-production environment, ensuring that any potential issues are identified before making adjustments to the live system.