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JEE Main202324 Jan 2023Evening ShiftMathematicsBinomial TheoremActual

If C 1 30 2 + 2 C 2 30 2 + 3 C 3 30 2 . . . . . . . . . . 30 C 30 30 2 = α 60 ! 30 ! 2 , then α is equal to

Options

  1. A30
  2. B60
  3. C15
  4. D10

Correct answer

C. 15

Step-by-step solution

Let, S = 0 C 1 30 2 + 1 C 1 30 2 + 2 C 2 30 2 + 3 C 3 30 2 . . . . . . . . . . 30 C 30 30 2 Now rewriting the above expression we get, S = 30 C 30 30 2 + 29 C 29 30 2 + 28 C 28 30 2 + 27 C 27 30 2 . . . . . . . . . . + 0 C 0 30 2 Now adding both the equation we get, 2 S = 30 C 1 30 2 + C 1 30 2 + C 2 30 2 + C 3 30 2 . . . . . . . . . . C 30 30 2 ⇒ S = 15 C 30 2 × 30 ⇒ S = 15 C 30 60 = 15 × 60 ! 30 ! 2 Now comparing with S = α 60 ! 30 ! 2 , We get α = 15

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