The correct statement among the given options is Option 3: The Prime Minister of India has full discretion in the choice of persons who are to serve as ministers in his cabinet.

In India`s parliamentary system, the Prime Minister is the head of the government and has significant powers and responsibilities. The process of forming a cabinet is an important aspect of governance. Let`s analyze each option and provide reasoning for why Option 3 is correct:

Option 1: "The Prime Minister is free to choose his ministers only from among those who are members of either House of the Parliament."

This statement is not entirely accurate. While it is true that ministers are typically chosen from among the members of either House of the Parliament, there is no legal restriction limiting the Prime Minister`s choice to only Parliament members. The Prime Minister has the authority to select individuals outside of Parliament as ministers if they are deemed suitable for the positions. However, it is often considered politically prudent for the Prime Minister to choose ministers who are members of Parliament, as they are accountable to the legislature.

Option 2: "The Prime Minister can choose his cabinet colleagues after due counseling by the President of India in this regard."

This statement is incorrect. The President of India does not have a role in counseling the Prime Minister regarding the selection of cabinet colleagues. The President`s role is largely ceremonial and symbolic, and the appointment of cabinet ministers falls within the Prime Minister`s domain.

Option 3: "The Prime Minister has full discretion in the choice of persons who are to serve as ministers in his cabinet."

This statement is accurate. The Prime Minister holds significant discretionary power in selecting individuals to serve as ministers in the cabinet. The Prime Minister can consider various factors such as competence, experience, party affiliations, and political considerations while choosing ministers. This authority allows the Prime Minister to build a team that aligns with the government`s vision and policies.

Option 4: "The Prime Minister has only limited powers in the choice of his cabinet colleagues because of the discretionary power vested with the President of India."

This statement is incorrect. As mentioned earlier, the President of India does not possess discretionary power over the choice of cabinet colleagues. The President`s role is largely ceremonial, and the Prime Minister has the primary authority in selecting and appointing ministers.

In conclusion, Option 3 is the correct statement. The Prime Minister of India has full discretion in the choice of persons who are to serve as ministers in his cabinet. This discretionary power allows the Prime Minister to form a team that best suits the government`s objectives and policies.