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JEE Main202622 January 2026Morning ShiftMathematicsBinomial TheoremActual

The coefficient of x⁴⁸ in (1+x)+2(1+x)²+3(1+x)³+ +100(1+x)¹⁰⁰ is equal to

Options

  1. A100 ¹⁰⁰ C ₄₉- ¹⁰⁰ C ₄₈
  2. B¹⁰⁰ C ₅₀+ ¹⁰¹ C ₄₉
  3. C^ 100 ¹⁰¹ C ₄₉- ¹⁰¹ C ₅₀
  4. D100 ¹⁰⁰ C ₄₉- ¹⁰⁰ C ₅₀

Correct answer

C. ^ 100 ¹⁰¹ C ₄₉- ¹⁰¹ C ₅₀

Step-by-step solution

Let 1 + x = r S = 1 r + 2 r^2 + 3 r^3 + + 100r¹⁰⁰ ......(1) (AGP) rS = 1 r^2 + 2 r^3 + + 99r¹⁰⁰ + 100r¹⁰¹ ......(2) (1) - (2) gives S = - (1+x)¹⁰¹ x^2 + 1 x^2 + 100(1+x)¹⁰¹ x coefficient x⁴⁸ in S = - coefficient of x⁴⁸ in (1+x)¹⁰¹ x^2 + 100 Coefficient of x⁴⁸ in (1+x)¹⁰¹ x = 100 101 49 - 101 50

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