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NTA Abhyas JEE Main2020MathematicsBinomial TheoremPractice

Let C 1 ,   C 2 ,   C 3 . . . are the usual binomial coefficients where C r =   n C r . Let S = C 1 + 2 C 2 + 3 C 3 + . . . + n C n , then S is equal to

Options

  1. An 2 n
  2. B2 n - 1
  3. Cn 2 n - 1
  4. D2 n + 1

Correct answer

C. n 2 n - 1

Step-by-step solution

Let S = C 1 + 2 C 2 + 3 C 3 + … + n C n = Σ r = 1 n  r ·   n C r = Σ r = 1 n  r · n r   n - 1 C r - 1       ∵   n C r = n r   n - 1 C r - 1 = n Σ r = 1 n   n - 1 C r - 1 = n   n - 1 C 0 +   n - 1 C 1 +   n - 1 C 2 + … +   n - 1 C n - 1 = n 2 n - 1

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