# Sixteen players S_(1), S_(2), S_(3),…,S_(16) play in a tournament. Number of ways in which they can be grouped into eight pairs so that S_(1)

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Sixteen players S_(1), S_(2), S_(3),…,S_(16) play in a tournament. Number of ways in which they can be grouped into eight pairs so that S_(1) and S_(2) are in different groups, is equal to
A. ((14)!)/(2^(6)*6!)
B. ((15)!)/(2^(7)*7!)
C. ((14)!)/(2^(7)*6!)
D. ((14)!)/(2^(6)*7!)

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(a) Required number of ways
=("Number of ways in which" 16 "players can be divided in" 8 "couples")-("Number of ways when" S_(1) "and" S_(2) "are in the same group")
=(16!)/(2^(8)*8!)-((14)!)/(2^(7)*7!)=(16*15*14!)/(2*2^(7)*8*7!)-((14)!)/(2^(7)*7!)
=((14!))/(2^(7)*7!)[(16*15)/(16)-1]=((14)*(14)!)/(14*(6!)*2^(6))=((14!))/(2^(6)*6!)`