Final answer:
Brokers in the state of North Carolina cannot operate independently or designate themselves as broker in charge.
Step-by-step explanation:
In the state of North Carolina, brokers are required to work under a broker in charge. This means that they do not have the ability to operate independently (Option A is false). Additionally, brokers in North Carolina cannot designate themselves as broker in charge. They must work under a designated broker in charge (Option C is false). Therefore, the correct statement is D. Both A and C are false.