Vault 21

 is one of the Vault-Tec Vaults, located in or near Las Vegas, Nevada. Its purpose in the Vault Experiment was gambling, which was confirmed in the June 2010 Issue of Game Informer. "The Vault 21 experiment was that it was the gambling vault. All conflicts would be resolved through gambling." Mitchell comes from there.

Appearances
A Vault 21 jumpsuit appears in Fallout: New Vegas, so it's very likely that so does the vault itself. Doctor Mitchell that heals you in the beginning gives his old vault suit to you instead of your old ruined clothes he also gives you his old Pip-Boy 3000 (reference: Xbox Magazine, April 2010).