In him we were also chosen , having been predestined according to the plan of him who works out everything in conformity with the purpose of his will, in order that we, who were the first to hope in Christ , might be for the praise of his glory. And you also were included in Christ when you heard the word of truth, the gospel of your salvation. The choosing applies to those in Christ. Predestination has to do with redemption and forgiveness of sins and adoption as sons of God in Christ. The predestined are those who are in Christ.
Eternal life is open to all who believe.
For God sent his Son into the world not to judge the world, but so that through him the world might be saved. In Him we were chosen. Paul could just as easily said, Having believed you were chosen. He could just as easily said, Having believed you were predestined. Christ did all of that for us.