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Manipal MET2020MathematicsBinomial Theorem

The coefficient of x^4 in (1+x+x^3+x^4 )¹⁰ is

Options

  1. A210
  2. B100
  3. C310
  4. D110

Correct answer

C. 310

Step-by-step solution

Let E= (1+x+x^3+x^4 )¹⁰= [1+x+x^3(1+x) ]¹⁰=(1+x)¹⁰ (1+x^3 )¹⁰ aligned = [ ¹⁰ C₀+ ¹⁰ C₁ x^1 . & + ¹⁰ C₂ x^2+ + ¹⁰ C₉ x^9 & .+ 10 10 x¹⁰ ] [¹⁰ C₀ 1+ ¹⁰ C₁ 1 (x^3 ) . & .+ ¹⁰ C₂ (x^3 )^2+ ¹⁰ C₃ (x^3 )^3+ ] aligned Coefficient of x^4 in E= ¹⁰ C₄ ¹⁰ C₀+ ¹⁰ C₁ ¹⁰ C₁= 10! 6! 4! 1+10 10= 10 9 8 7 4 3 2 1 +100=210+100=310

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