As in the colorful expression "I don't give a rat's ass." The origin of this phrase is unknown. Perhaps it came from "I don't have a care in the world," which is an interesting use of the word care, which is usually a verb. A more mild version is "I don't give a flip." Also "I don't give a shit." Most people don't give their shit to anyone except doctors, and even then only when explicitly asked for. Giving a flying fuck would be rather difficult even if you were equiped with a twin-person hang glider. The weight shifts would probably send you out of control.