Hat check problem. Ten friends total, five with sombreros, five with fedoras.

Question

A group of ten people give their hats to the coatroom attendant. Five of the ten are wearing sombreros, and five and wearing fedoras. How many ways can the clerk return the hats so that no one gets their hat back if,

a: No one gets the right kind of hat

b: Everyone gets the right kind of hat

c: The attendant loses the sombreros and only returns the 5 fedoras. However the five people in the group that do get a hat back, may or may not have gotten the correct hat back.

Answer 1

a. 5!*5! If the kind of hat is wrong, we just get permutations.

b. D5 * D5 (Where D5 is the number of derangements on 5 objects = [5!/e]) This derangement number is determined with the inclusion/exclusion method.

c. 10*9*8*7*6. Without the correct hat/person limitations, the attendant can give the five fedoras to 10, 9, 8, 7, 6 persons respectively (assuming every person gets one hat)