The Prime Minister (PM) and Chief Minister (CM) are elected through the parliamentary system in India. The parliamentary system is a democratic form of government where the executive branch is accountable to the legislature.

**Election of the Prime Minister:**
1. General Elections: The first step in electing the Prime Minister is the general elections. These elections are conducted across the country to elect Members of Parliament (MPs) to the Lok Sabha, the lower house of the Parliament.
2. Formation of Government: After the general elections, the political party or coalition that secures a majority of seats in the Lok Sabha forms the government. The leader of the majority party becomes the Prime Minister.
3. Prime Minister's Appointment: The President of India, who is the head of the state, appoints the Prime Minister. The President's role is primarily ceremonial, and the appointment is made based on the majority support in the Lok Sabha.
4. Swearing-in Ceremony: Once appointed, the Prime Minister takes the oath of office and secrecy before the President. The Prime Minister then forms the Council of Ministers, who are responsible for assisting in the governance of the country.

**Election of the Chief Minister:**
1. State Legislative Assembly Elections: The election of the Chief Minister happens at the state level. State Legislative Assembly elections are conducted to elect Members of Legislative Assembly (MLAs) to the respective state assemblies.
2. Formation of Government: Similar to the national level, the political party or coalition that secures a majority of seats in the state assembly forms the government. The leader of the majority party becomes the Chief Minister.
3. Chief Minister's Appointment: The Governor of the state, who is the head of the state, appoints the Chief Minister. The Governor's role is similar to that of the President at the national level, and the appointment is made based on the majority support in the state assembly.
4. Swearing-in Ceremony: Once appointed, the Chief Minister takes the oath of office and secrecy before the Governor. The Chief Minister then forms the Council of Ministers, who are responsible for assisting in the governance of the state.

The election of the PM and CM through the parliamentary system ensures that the executive branch is accountable to the legislature. The PM and CM are elected by the representatives of the people, who are elected through a democratic process. This system promotes representative democracy and allows for the selection of leaders who have the support of the majority in the legislature.