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NDA2025MathematicsBinomial TheoremActual

Which one of the following is the greatest coefficient in the expansion of (1 + x)¹⁰⁰ ?

Options

  1. AThe coefficient of x¹⁰⁰
  2. BThe coefficient of x⁹⁹
  3. CThe coefficient of x⁵¹
  4. DThe coefficient of x⁵⁰

Correct answer

D. The coefficient of x⁵⁰

Step-by-step solution

The general term in the expansion of (1 + x)^ n is given by T_ r+1 = ^ n C_ r x^ r , where the coefficient of x^ r is ^ n C_ r . For an even integer n , the greatest binomial coefficient is the middle term, which is ^ n C_ n/2 . Given n = 100 , which is even, the greatest coefficient is ¹⁰⁰C_ 100/2 = ¹⁰⁰C₅₀ . This corresponds to the coefficient of x⁵⁰ . Answer: The coefficient of x⁵⁰

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