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NTA Abhyas JEE Main2020MathematicsBinomial TheoremPractice

The coefficient of x n - 2 in the polynomial x - 1 x - 2 x - 3 … … … x - n is

Options

  1. An n 2 + 2 3 n + 1 24
  2. Bn n 2 - 1 3 n + 2 24
  3. Cn n 2 + 1 3 n + 4 24
  4. Dn n 2 - 2 3 n - 2 24

Correct answer

B. n n 2 - 1 3 n + 2 24

Step-by-step solution

Let, E = x - α 1 x - α 2 x - α 3 … x - α n where α 1 = 1 , α 2 = 2 etc. = x n - ∑ α 1 x n - 1 + ∑ α 1 α 2 x n - 2 + . . . Hence, the coefficient of x n - 2 = the sum of all the products of the first ‘ n ’ natural numbers taken two at a time 1 + 2 + 3 + . … … + n 2 - 1 2 + 2 2 + 3 2 + . … … . + n 2 2 = n n 2 - 1 3 n + 2 24

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