How do India elect its prime …

There is no direct election to the post of the Prime Minister. The President appoints the Prime Minister. But the President cannot appoint anyone he likes. He appoints the leader of the majority party or the coalition of parties that commands a majority in the Lok Sabha, as Prime Minister. In case no single party or alliance gets a majority, the President appoints the person most likely to secure a majority support. The Prime Minister does not have a fixed tenure. He continues in power so long as he remains the leader of the majority party or coalition. 
Since the Prime Minister is the head of the government, he enjoys wide-ranging powers : 
(i) He chairs Cabinet meetings and coordinates the work of different Departments. His decisions are final in case disagreements arise between Departments. 
(ii) He exercises general supervision of different ministries. All ministers work under his leadership. 
(iii) The Prime Minister distributes and redistributes work to the ministers. 
(iv) He also has the power to dismiss ministers. When the Prime Minister quits, the entire ministry quits. 
Thus, the Prime Minister is the most powerful person within the Cabinet.