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JEE Main20252 Apr 2025Evening ShiftMathematicsBinomial TheoremActual

If _ r =0 ¹⁰ ( 10^ r +1 -1 10^ r ) ¹¹ C _ r +1 = ¹¹-11¹¹ 10¹⁰ , then is equal to :

Options

  1. A15
  2. B11
  3. C24
  4. D20

Correct answer

D. 20

Step-by-step solution

aligned & _ r =0 ¹⁰ ( 1 r ^ r -1 -1 10^ r ) ¹¹ C _ r +1 & = _ r =0 ¹⁰ (10- 1 10^ r )¹¹ C _ r +1 & =10 _ r =0 ¹⁰ C _ r +1 -10 ( ¹¹ C _ r +1 ( 1 10 )^ r +1 ) & =10 [ ¹¹ C ₁+ ¹¹ C ₂+ . .+ ¹¹ C ₁₁ ] & -10 [ ¹¹ C ₁ ( 1 10 )^1+ ¹¹ C ₂ ( 1 10 )^2+ . .+ ¹¹ C ₁₁ ( 1 10 )¹¹ ] & =10 [2¹¹-1 ]-10 [ (1+ 1 10 )¹¹-1 ] & =10(2)¹¹-10- 11¹¹ 10¹⁰ +10 & = (20)¹¹-11¹¹ 10¹⁰ & =20 aligned

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