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Question

The number of ways in which 100 people can be divided in 50 pairs is


A
100!250(50!)
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B
100!250
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C
50n=26  2nC22
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D
50n=1 2nC2
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Solution

The correct options are
A 100!250(50!)
C 50n=26  2nC22
Number of ways of arranging 100 people is 100!
But, when we make a pair then both the persons in a pair are considered as one person. So, we divide it by 2!×2!×2!upto 50 times. 
So, we get 100!250 ways.
But, the above case considers the arrangement of 50 couples also which should be eliminated.
So, the required number of ways is 100!250×50!

Also,

100!250×50!=512×522×532×...×1002
=50n=26 2nC22

Mathematics

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