What does Romans 11:5 mean?
Paul is making the case that God has not rejected Israel as His people, as well as the fact that a remnant of Israel has remained faithful to God by faith in Christ. This small portion of the Jewish people have received Christ as Messiah instead of rejecting Him in order to try to be saved by their works through the law.Paul proclaims now that this remnant has been chosen by God's grace. By this, Paul means that they have not earned their place in this faithful remnant of Israelites. Just as Gentile Christians are chosen by God's grace, these Israelites have received salvation as a gift. Paul wants to be clear they have not earned it in any way by being faithful to the law of Moses.
The following verse not only summarizes this point, it also stands alone as a crucial part of Christian theology. Grace cannot be mixed with good works—either salvation is attained according to God's grace, or it's earned by deeds. It cannot be both at the same time. Paul has already demonstrated that salvation by one's own works is impossible (Romans 3:10, 20).