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JEE Main202228 Jun 2022Morning ShiftMathematicsBinomial TheoremActual

If ∑ k = 1 31 C k 31 C k - 1 31 - ∑ k = 1 30 C k 30 C k - 1 30 = α 60 ! 30 ! 31 ! , where α ∈ R , then the value of 16 α is equal to

Options

  1. A1411
  2. B1320
  3. C1615
  4. D1855

Correct answer

A. 1411

Step-by-step solution

∑ k = 1 31 C k 31 · C k - 1 31 = C 1 31 · C 0 31 + C 2 31 · C 1 31 + ⋯ + C 31 31 · C 30 31 = C 0 31 · C 30 31 + C 1 31 · C 29 31 + ⋯ + C 30 31 · C 0 31 (using C r n = C r - 1 n ) = C 30 62 (using C 0 n · C p m + C 1 n · C p - 1 m + ⋯ + C p n · C 0 m = C p m + n ) Similarly, ∑ k = 1 30 C k 30 · C k - 1 30 = C 29 60 C 30 62 - C 29 60 = 62 ! 30 ! 32 ! - 60 ! 29 ! 31 ! = 60 ! 29 ! 31 ! 62 · 61 30 · 32 - 1 = 60 ! 30 ! 31 ! 282

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