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Question

The value of 11!(n1)!+13!(n3)!++1(n1)!(1)!, where n is even positive integer, is
  1. 2n(n1)!
  2. 2n1n!
  3. 2n+1n!
  4. 2n(n+1)! 


Solution

The correct option is B 2n1n!
Let
S=11!(n1)!+13!(n3)!++1(n1)!(1)!S=1n![n!1!(n1)!+n!3!(n3)!++n!(n1)!(1)!]S=1n![ nC1+ nC3++ nCn1]S=2n1n!

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