The correct option is 2 and 3 only.
Explanation
The office of the Chief Minister is established under Article 163 and Article 164 of the Constitution of India. The Chief Minister is the head of the Council of Ministers at the State level and occupies a position in the State executive analogous to that of the Prime Minister at the Union level.
- Statement 1 — The Chief Minister has the power to appoint the Governor of the State: Incorrect
Under Article 155 of the Constitution, the Governor of a State is appointed by the President of India by warrant under his hand and seal. The Chief Minister has no constitutional role or power in the appointment of the Governor. The Governor is neither elected nor appointed on the advice of the Chief Minister. This power is exclusively vested in the President of India, making Statement 1 constitutionally incorrect. - Statement 2 — The Chief Minister has the responsibility of running government departments: Correct
The Chief Minister, as the head of the Council of Ministers, is responsible for the overall administration and functioning of State government departments. Under Article 166(3), the Governor makes rules for the more convenient transaction of the business of the State Government and for the allocation of business among Ministers — but it is the Chief Minister who coordinates and oversees the actual running of these departments. The Chief Minister allocates portfolios to Ministers and is accountable for the administration of the State. - Statement 3 — The Chief Minister is the head of the Executive part of the state government: Correct
While the Governor is the constitutional head of the State, the Chief Minister is the real executive head of the State Government. Under the parliamentary system adopted by the Indian Constitution, executive power is exercised by the Council of Ministers headed by the Chief Minister. The Governor acts on the aid and advice of the Council of Ministers (as per Article 163), making the Chief Minister the de facto head of the State executive.
Key Takeaway: The Governor of a State is appointed by the President of India under Article 155 — the Chief Minister has no role in this appointment; the Chief Minister is the real/de facto executive head of the State government, responsible for running government departments and coordinating the Council of Ministers.