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JEE Main2016MathematicsBinomial TheoremActual

For x ∈ R , x ≠ - 1 , if 1 + x 2016 + x 1 + x 2015 + x 2 1 + x 2014 + … + x 2016 = ∑ i = 0 2016 a i x i , then a 17 is equal to

Options

  1. A2017 ! 17 ! 2000 !
  2. B2016 ! 17 !1 999 !
  3. C2016 ! 16 !
  4. D2017 ! 2000 !

Correct answer

A. 2017 ! 17 ! 2000 !

Step-by-step solution

S = 1 + x 2016 + x 1 + x 2015 + x 2 1 + x 2014 + … … + x 2015 1 + x + x 2016     . . . i x 1 + x   S = x   1 + x 2015 + x 2 1 + x 2014 + … + x 2016 + x 2017 1 + x     . . . i i i i - i S 1 + x = 1 + x 2016 - x 2017 1 + x ∴ S = 1 + x 2017 - x 2017 Coefficient a 17 = 2017 C 17   = 2017 ! 17 ! 2000 !

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