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AP EAMCET201726 Apr 2017Morning ShiftMathematicsBinomial TheoremActual

The coefficient of x^5 in the expansio of (1+x)²¹+(1+x)²²+ +(1+x)³⁰ is

Options

  1. A³¹ C ₆- ²¹ C ₆
  2. B⁵¹ C₅
  3. C^9 C ₅
  4. D³⁰ C ₅+ ²⁰ C ₅

Correct answer

A. ³¹ C ₆- ²¹ C ₆

Step-by-step solution

As we know, coefficient of x^r in the binomial expansion of (1+x)^n is given by ^n C_r . So, coefficient of x^5 in the binomial expansion of aligned & (1+x)²¹+(1+x)²²+ .+(1+x)³⁰ & = ²¹ C₅+ ²² C₅+ .+ ³⁰ C₅ & = ( ²¹ C₆+ ²¹ C₅+ ²² C₅+ .+ ³⁰ C₅ )- ²¹ C₆ & = ( ²² C₆+ ²² C₅+ .+ ³⁰ C₅ )- ²¹ C₆ & [ ^n C_r+ ^n C_ r-1 = ^ n+1 C_r ] & = ( ²³ C₆+ ²³ C₆+ .+ ³⁰ C₅ )- ²¹ C₆ aligned = ( ³⁰ C₆+ ³⁰ C₅ )- ²¹ C₆= ³¹ C₆- ²¹ C₆

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