The correct answer is option D: 1 and 5.

1. National Integration Council:
The National Integration Council (NIC) is a non-constitutional mechanism that promotes coordination between the center and states in matters related to national integration. It is a consultative body that discusses and suggests measures to address issues pertaining to communal harmony, national security, and socio-economic development. The NIC consists of members representing the central government, state governments, political parties, and various social and religious organizations.

2. Finance Commission:
The Finance Commission is a constitutional body that plays a crucial role in promoting coordination between the center and states in fiscal matters. It is constituted every five years and its primary function is to recommend the distribution of financial resources between the center and states. The Finance Commission determines the share of taxes and grants-in-aid to be allocated to states, taking into account various factors such as population, area, fiscal capacity, and developmental needs of the states.

3. Regional Development Council:
Unlike the National Integration Council, the Regional Development Council (RDC) is not a formal body established by the constitution or law. However, it is a non-constitutional mechanism that promotes coordination and cooperation between the center and states in matters of regional development. RDCs are set up for specific regions and serve as platforms for dialogue and consultation between the central government, state governments, and other stakeholders to coordinate efforts and resources for the development of the region.

4. National Advisory Council:
The National Advisory Council (NAC) was a non-constitutional body that was constituted by the government to provide inputs and recommendations on various policy issues. It played a significant role in promoting coordination between the center and states by facilitating dialogue and consultation between the government and civil society organizations. However, the NAC was disbanded in 2014.

5. University Grants Commission:
The University Grants Commission (UGC) is a statutory body established by an Act of Parliament. It is responsible for the coordination, determination, and maintenance of standards of higher education in India. While the UGC does not directly promote coordination between the center and states, it plays a crucial role in promoting uniformity and coordination among universities across the country by formulating regulations and guidelines for higher education institutions.

In conclusion, among the given options, the National Integration Council and the University Grants Commission are the non-constitutional mechanisms that promote coordination between the center and states. The Finance Commission is a constitutional body, and the National Advisory Council was disbanded in 2014. The Regional Development Council is not a formal body established by the constitution or law. Therefore, the correct answer is option D: 1 and 5.