What dramatic breakthrough allowed nomadic bands to settle in villages for the first time

About 10,000 years ago they made the breakthrough that was the skill of farming and to produce their own food so they could remain in one place.

Agriculture is answer you're looking for! Nomadic bands had to hunt or gather their food which meant they had to move to find it if the weather changed or if their wasn't anything to eat in the area. That all changed when someone figured out how to grow their own food so they wouldn't have to leave a spot. About 9,500 BC is when we first got evidence of that although we think that wild grains were collected and eaten from at least 20,000 BC.

The answer is option b and c.