A group of n people went to a theatre, and each of them left his coat and his suitcase in the...

Question

A group of n people went to a theatre, and each of them left his coat and his suitcase in the coatroom. When the group wanted to leave, the coatroom attendant handed each person a coat and a suitcase in a random fashion. (a) In how many ways could the attendant return the coats and suitcases if nobody got his own coat back and nobody got his own suitcase back? (b) In how many ways could the attendant return the coats and suitcases if some people might have gotten one of them belongings back, but nobody got two of his belongings back?

Komalavalli · Accepted Answer

a)
Total number of ways of distribution coats and suitcase = number of permutations of coats and suitcase is n!. Number of ways when no one gets his own coats and suitcase is calculated as follows: fix a person; he can choose among n-1 coats and suitcase, then, fix another person, he has n-2 coats and suitcase at his disposal, and so on. This leads to n-1!
Therefore attendant returns the coats and suitcases if nobody got his own coat back and nobody got his own suitcase back in (n-1)! Ways. 
b)
the attendant return the coats and suitcases if some people might have gotten one of them belongings back,

A group of n people went to a theatre, and each of them left his coat and his suitcase in the coatroom. When the group wanted to leave, the coatroom attendant handed each person a coat and a suitcase...

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