Once Frodo and Sam reawakened nearly two weeks after the fall of Sauron in Ithil

Question

Once Frodo and Sam reawakened nearly two weeks after the fall of Sauron in Ithilien, the eight remaining members of the Fellowship celebrated their long awaited reunion at a Gondorian tavern. After hours of mirth, the bartender failed to return the cloaks to their proper owners when they were ready to leave, although he was able to distinguish the two relative cloak sizes:

• The ?ve smaller ones of the four hobbits (Frodo, Sam, Merry, Pippin) and Gimli. • The three larger ones of Gandalf, Legolas, and Aragorn.

How many ways can this happen?

Explanation / Answer

Total number of desired ways= Derangement of 4 things X Derangement of 3 other things

DN= N![1-1/1!+1/2!-1/3!…1/N!]

Derangement of 4 cloaks = D4 = 9 ways

Derangement of 3 cloaks = D3 = 2 ways

Hence, total number of ways = 18 ways.

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