The preselected list of candidates is usually recommended by people in the President's political party or by members of legislation (House of Representatives, and Congress), usually it's a combination of both.
After a candidate has been selected, the Senate Judiciary Committee conducts hearings (questioning) on the candidate. The hearing is meant to determine whether the candidate is qualified and suitable for the position.
After the Committee reviews the nominee, they pass a recommendation to reject or confirm to the Senate floor. The Senate then votes for or against the candidate. In order to become a Supreme Court Justice, the nominee must receive a simple majority (51 votes) of the Senate, unless a group chooses to filibuster, in which case a 3/5ths super majority is required to complete the appointment.
It is highly unlikely that a candidate will be rejected. Since 1789, the Senate has rejected 30 out of the 144 nominees, the most recent being Robert Bork in 1987.
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