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JEE Main202126 Aug 2021Morning ShiftMathematicsBinomial TheoremActual

If C r 20 is the co-efficient of x r in the expansion of ( 1 + x ) 20 , then the value of ∑ r = 0 20 r 2 C r 20 is equal to:

Options

  1. A420 × 2 18
  2. B380 × 2 18
  3. C380 × 219
  4. D420 × 2 19

Correct answer

A. 420 × 2 18

Step-by-step solution

∑ r = 0 20 r 2 C r 20 = ∑ r r · C r 20 = ∑ r = 1 20 r · 20 . C r - 1 19 ⇒ 20 ∑ r = 1 20 ( r - 1 + 1 ) · 19 C r - 1 = 20 ∑ r = 1 20 ( r - 1 ) · 19 C r - 1 + ∑ r = 1 20 C r - 1 19 ⇒ 20 19 ∑ r = 2 20 C r - 2 18 + ∑ r = 1 20 C r - 1 19 ⇒ ( 380 ) 2 18 + 20 · 2 19 ⇒ 2 20 · ( 95 + 10 ) = ( 105 ) 2 20 = 420 × 2 18

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