Question

# Number of ways in which 200 people can be divided into 100 pairs is

A
(200!)2100(100)!
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B
1.3.5199
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C
(200!)(100)!
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D
(1012),(1022)(2002)
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Solution

## The correct options areA (200!)2100(100)!B. 1.3.5……199D (1012),(1022)……(2002)Firstly, number of ways in which 200 people can be arranged at 200 places is 200P200=200!……(1)Now, these 200 people have been divided into 100 couples and order of couples is immaterial.Thus, number of ways in which these 100 couples can be arranged at 100 places is=100P100=100!……(2) We have to divide (1) by (2) Again, each pair of couple can be arranged in two different ways as order in which couple is formed is immaterial.Thus, number of ways in which 100 couples can be arranged within themselves is, =[2×2×2×……]100 times =2100……(3)Thus, total number of ways in which 200 people can be arranged in 100 couples is =(200!)2100(100)!Therefore, required number of ways =(200!)2100(100)!=(100)!(101)(102)(103).......(200)2100(100!)=(1012)(1022)……(2002)So, Option D also a correct answer.And also(200!)2100(100)! is in the form of (2n)!2n(n!), Here n=100As we know that (2n)!2n(n!)=1⋅3⋅5⋅7⋅⋅⋅⋅⋅⋅(2n−1)Since n=100So, (200!)2100(100)!=1⋅3⋅5⋅7⋅⋅⋅⋅⋅⋅199Therefore, options A,B,D are correct answers.

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