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JEE Main202624 January 2026Morning ShiftMathematicsBinomial TheoremActual

Let S = 1 25! + 1 3!23! + 1 5!21! + up to 13 terms. If 13 ~S = 2^ k n! , k ~N , then n+k is equal to

Options

  1. A50
  2. B49
  3. C52
  4. D51

Correct answer

B. 49

Step-by-step solution

The general term is 1 (2r-1)!(27-2r)! for r=1 to 13 . Multiply and divide by 26! : 1 (2r-1)!(27-2r)! = 1 26! 26 2r-1 S = 1 26! _ r=1 ¹³ 26 2r-1 = 1 26! [ 26 1 + 26 3 + + 26 25 ] Sum of odd-indexed binomial coefficients: 26 odd = 2²⁵ S = 2²⁵ 26! 13S = 13 2²⁵ 26! = 13 2²⁵ 26 25! = 2²⁵ 2 25! = 2²⁴ 25! So k = 24, ; n = 25 , giving n + k = 49 .

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