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JEE Main2012MathematicsBinomial TheoremActual

If f(y)=1-(y-1)+(y-1)^2-(y-1)^3+ -(y-1)¹⁷ then the coefficient of y^2 in it is

Options

  1. A¹⁷ C ₂
  2. B¹⁷ C ₃
  3. C¹⁸ C ₂
  4. D¹⁸ C ₃

Correct answer

D. ¹⁸ C ₃

Step-by-step solution

Given function is array r f(y)=1-(y-1)+(y-1)^2-(y-1)^3 + -(y-1)¹⁷ array In the expansion of (y-1)^n T_ r+1 = ^n C _r y^ n-r (-1)^r coeff of y^2 in (y-1)^2= ^2 C ₀ coeff of y^2 in (y-1)^3= ^3 C ₁ coeff of y^2 in (y-1)^4= ^4 C ₂ So, coeff of termwise is aligned & ^2 C ₀+ ^3 C ₁+ ^4 C ₂+ ^5 C ₃+ + ¹⁷ C ₁₅ & =1+ ^3 C ₁+ ^4 C ₂+ ^5 C ₃+ . .+ ¹⁷ C ₁₅ & = ( ^3 C ₀+ ^3 C ₁ )+ ^4 C ₂+ ^5 C ₃+ . .+ ¹⁷ C ₁₅ & = ^4 C ₁+ ^4 C ₂+ ^5 C ₃+ + ¹⁷ C ₁₅ aligned aligned & = ^5 C ₂+ ^5 C ₃+ + ¹⁷ C ₁₅ & = ¹⁸ C ₁₅= ¹⁸ C ₃ aligned

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