2. Once Frodo and Sam reawakened nearly two weeks after the fall of Sauron in It

Question

2. 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 five 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

So every cloak is returned to the wrong person. This happened in two different cases - the long cloak case and the short cloak case.

D(5) - this means that none of the five people got their own cloak. Similar for D(3)

D(5) = 5! * [ 1- 1/1! + 1/2! - 1/3! + 1/4! -1/5!]

D(3) = 3! * [ 1- 1/1! + 1/2! - 1/3! ]

final ans = D(5) * D(3) = 88

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