There are 19 US Post Offices in Buncombe County serving a total of 250,198 residents equating to 13,168 residents per post office. It's estimated that approximately 399,344 packages are handled across the 19 county post offices annually. Buncombe County is a county located in the U.S. state of North Carolina. As of the 2010 census, the population was 238,318. Its county seat is Asheville. Buncombe County is part of the Asheville, NC Metropolitan Statistical Area.