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NTA Abhyas JEE Main2020MathematicsBinomial TheoremPractice

∑ r = 0 n r n - r n C r 2 is equal to

Options

  1. An 2 2 n - 1 C n
  2. Bn 2 2 n C n - 1
  3. Cn 2 2 n - 1 C n - 1
  4. Dn 2 2 n - 2 C n

Correct answer

D. n 2 2 n - 2 C n

Step-by-step solution

∑ r = 0 n r ⋅ n C r n - r n C n - r = ∑ r = 1 n - 1 n ⋅ n - 1 C r - 1 n · n - 1 C n - r - 1 = n 2 ∑ r = 1 n - 1   n - 1 C r - 1 · n - 1 C n - r - 1 = n 2 ∑ r = 1 n - 1 n - 1 C r - 1 n - 1 C r = n 2 n - 1 C 0 n - 1 C 1 + n - 1 C 1 n - 1 C 2 + . . . . . + n - 1 C n - 2 n - 1 C n - 1 = n 2 × coefficient of x n - 2 in 1 + x n - 1 1 + x n - 1 = n 2 2 n - 2 C n - 2 = n 2 2 n - 2 C n

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