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AP EAMCET201823 Apr 2018Evening ShiftMathematicsBinomial TheoremActual

The coefficient of x⁵⁰ in the expansion of (1+x)¹⁰⁰+2 x(1+x)⁹⁹+3 x^2(1+x)⁹⁸++101 x¹⁰⁰ , is

Options

  1. A¹⁰⁰ C ₅₀
  2. B¹⁰¹ C ₅₀
  3. C¹⁰² C ₅₀
  4. D¹⁰³ C ₅₀

Correct answer

C. ¹⁰² C ₅₀

Step-by-step solution

aligned & Let S=(1+x)¹⁰⁰+2 x(1+x)⁹⁹+3 x^2(1+x)⁹⁸ & + +101 x¹⁰⁰ & x 1+x S= & x(1+x)⁹⁹+2 x^2(1+x)⁹⁸+ +100 x¹⁰⁰+101 x¹⁰¹ 1+x & S 1+x =(1+x)¹⁰⁰+x(1+x)⁹⁹+x^2(1+x)⁹⁸ & + +x¹⁰⁰-101 x¹⁰¹ 1+x & = (1+x)¹⁰⁰ [ ( x 1+x )¹⁰¹-1 ] x 1+x -1 -101 x¹⁰¹ 1+x & S=(1+x)¹⁰²-x¹⁰²-102 x¹⁰¹ & aligned So, coefficient of x⁵⁰ in the expansion of S= 102 50 .

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