The correct answer is option 4: if he is convinced that the government of the State cannot be carried on in accordance with the provisions of the Constitution of India.

This means that the Governor of a state can recommend imposing President`s rule if he believes that the existing state government is unable to function according to the provisions of the Indian Constitution. President`s rule, also known as the central rule, is a situation where the President of India assumes direct control of the state government.

Option 1, on the recommendation of the State legislature, is incorrect because the Governor has the power to recommend President`s rule independently and doesn`t require the recommendation of the State legislature.

Option 2, on the recommendation of the President, is also incorrect because President`s rule is recommended by the Governor, not the President of India.

Option 3, on the recommendation of the Chief Minister, is also incorrect because the Chief Minister does not have the power to recommend the imposition of President`s rule in the state.

It is important to note that the decision to impose President`s rule is ultimately taken by the President of India based on the Governor`s recommendation.