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COMEDK2022MathematicsBinomial Theorem

The coefficient of x²⁰ in the expansion of (1+3 x+3 x^2+x^3 )²⁰ is

Options

  1. A60 40
  2. B30 20
  3. C¹⁵ C ₂
  4. DNone of these

Correct answer

A. 60 40

Step-by-step solution

The given expression is (1 + 3x + 3x^2 + x^3)²⁰ . Observe that 1 + 3x + 3x^2 + x^3 = (1 + x)^3 . Substituting this into the expression, we get ((1 + x)^3)²⁰ = (1 + x)⁶⁰ . The general term in the binomial expansion of (1 + x)⁶⁰ is given by ⁶⁰C_ r x^ r . To find the coefficient of x²⁰ , we set r = 20 . The coefficient is ⁶⁰C₂₀ . Using the property ^ n C_ r = ^ n C_ n-r , we have ⁶⁰C₂₀ = ⁶⁰C_ 60-20 = ⁶⁰C₄₀ . Answer: 60 40

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