Well both pre-destination and free will do co-exist according to the scriptures, God Is not culpable for the will of man which is sinful. Let me attempt again to show you pre-ordainment. Pre-ordain, means God does choose his elect, has chosen them from before creation. He gave all men the ability to choose him, but they will not do so. Men in their sin nature hate God and therefore God himself must choose men or none will be saved. That is about the best I can do right now after giving much thought how to put this into words that are not in scripture.
As far as the Genesis record, God for his own Glory chose before he even created earth, his people, He knew ahead of time his created would sin, Christ's sacrifice was already ordained. So despite what God knew would happen, He chose this way to bring glory to himself. As far as the individual sins of his created, God did not cause Adam and eve to sin, they did have a choice, they did disobey God there by sin entered the world thru one man. This inherent sin spoken of is not Biblical, It is Inherent sin nature, not responsibility of our fathers sin. Scripture tells us we are not punished for our fathers sin but that of our own, period.Once that apple was eaten, men were cursed with the sin nature, no man is Good, no not one. This is my belief and I believe this is what the Bible teaches us.