Four hats returned with at least one correct

Question

Four people hang up their hats, and the hats are returned at random, one to each person. In how many ways does at least one person get their own hat back?

Accepted Answer

"At least one correct" is the complement of "none correct," so subtract the derangements from the total. Total ways to return 4 hats: 4! = 24. Ways with no one matched (a derangement of 4): D4 = 9. At least one match: 4! - D4 = 24 - 9 = 15. Takeaway. "At least one fixed point" equals total minus derangements: n! - Dn. Here 24 - 9 = 15.

Four hats returned with at least one correct

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