Who appoints the prime minister of India?

• The prime minister of India is appointed by the President of India. Article 75 says that the Prime Minister shall be appointed by the President and the other Ministers shall be appointed by the President on the advice of the Prime Minister. 

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• A member of either the Lok Sabha or the Rajya Sabha can be appointed as the Prime Minister. If he is not a member of either House of the Parliament then he has to be elected to either House within six months of his appointment. 
• Following the conventions of the parliamentary system of government, the President has to appoint the leader of the majority party in the Lok Sabha as the Prime Minister. 
• But, when no party has a clear majority in the Lok Sabha, then the President may exercise his discretion in the selection and appointment of the Prime Minister. 
• In such a situation, the President usually appoints the leader of the largest party or coalition in the Lok Sabha as the Prime Minister and asks him to seek a vote of confidence in the House within a month.

Further Reading:

• Prime Minister
• Indian Polity Notes