The correct answer is option 3, the National Development Council. The Five-Year Plans in India are approved by the National Development Council (NDC). The NDC is a body that consists of the Prime Minister, Cabinet Ministers, Chief Ministers of all states, and members of the Planning Commission. The NDC is responsible for formulating and overseeing the implementation of economic and development policies in India.

Options 1, 2, and 4 are not the correct answer because they do not have the authority to approve the Five-Year Plans. The Union Cabinet, as the executive branch of the government, does play a crucial role in policy-making and decision-making processes. However, it does not have the sole authority to approve the Five-Year Plans. Similarly, the President on the advice of the Prime Minister does not have the authority to approve the plans independently. The Planning Commission, while involved in the formulation and implementation of the plans, does not have the final approval authority.

Therefore, option 3, the National Development Council, is the correct body responsible for approving the Five-Year Plans in India.