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JEE MainMathematicsBinomial Theorem

If the relation ^ n C_ r = ( ₃(x) - 1) ^ n+1 C_ r+1 holds for some integers n r 0 , then x must belong to the interval :

Options

  1. A(0, 9]
  2. B(3, 9]
  3. C(3, 9)
  4. D[3, 9]

Correct answer

B. (3, 9]

Step-by-step solution

Given the relation: ^ n C_ r = ( ₃(x) - 1) ^ n+1 C_ r+1 We can rewrite this as: ₃(x) - 1 = ^ n C_ r ^ n+1 C_ r+1 Expanding the combinations into factorials: ^ n C_r ^ n+1 C_ r+1 = n! r!(n-r)! (n+1)! (r+1)!(n-r)! = r+1 n+1 Thus, we have: ₃(x) - 1 = r+1 n+1 For the combinations to be valid, the integer r must satisfy 0 r n . Adding 1 to all parts of the inequality gives 1 r+1 n+1 . Dividing by n+1 (which is strictly positive), we get: 1 n+1 r+1 n+1 1 Since n can be any arbitrarily large integer, the lower bound appro

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