The correct answer is: c) Semi-autonomous entities.
A protectorate is a state that is formally independent but is under the protection of another state. A tributary state is a state that pays tribute to another state in exchange for protection. A direct administrative unit is a state that is directly controlled by another state.
Some princely states continued to exist under British rule as semi-autonomous entities. This means that they were not directly controlled by the British government, but they were still under British influence. The British government had the right to interfere in the affairs of the princely states, but they did not have the right to abolish them.
The princely states were important to the British Empire because they provided a buffer between British India and the rest of Asia. They also provided a source of revenue for the British government.
The princely states were abolished in 1947, when India gained independence from the British Empire.