Glove Exchange
There are five distinguishable pairs of gloves to be given to 5 persons. Each person must get a left glove and a right glove . Find the number of distributions so that no person gets a proper pair
The answer is 5280.
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2 solutions
My approach was
First all the cases where left glove is dearranged is to considered .There are 5 ! × ! 5 = 5 2 8 0 arrangements
Next all the cases where right gloves are dearranged is to be considered there are 5 ! × ! 5 = 5 2 8 0 arrangements.
Now the arrangements whee both pair of gloves are dearranged has to eliminated. So final answer - 2 × 5 2 8 0 − ! 5 2 = 8 6 2 4
First consider a mapping between left gloves and right gloves f : G i L → G i R where L and R denotes left and right of i t h gloves 1 ≤ i ≤ 5
If no pairs will be proper which means de-arrangement of the G i R .So , number of de-arrangement of 5 items will be 4 4 ways..
But there are 5 persons each person can get any one of the improper pairs.So G i L can arrange in 5 ! = 1 2 0 ways.
Total number = 4 4 × 1 2 0 = 5 2 8 0
For the left glove, there are 5 ! permutations.
For the right glove, there are ! 5 permutations.
Hence, the number of distributions so that no person gets a proper pair is
5 ! × ! 5 = 5 2 8 0
n ! states the number of permutations of n objects. You can read more about factorials .
! n states the number of permutations of n objects so that none of them stays in their original position. You can read more about derangements .