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Manipal MET2011MathematicsBinomial Theorem

The coefficient of x^4 in the expansion of (1+x+x^2+x^3 )^n is

Options

  1. A^n C₄
  2. B^n C₄+ ^n C₂
  3. C^n C₄+ ^n C₂+ ^n C₄ ^n C₂
  4. D^n C₄+ ^n C₂+ ^n C₁ ^n C₂

Correct answer

D. ^n C₄+ ^n C₂+ ^n C₁ ^n C₂

Step-by-step solution

aligned & (1+x+x^2+x^3 )^n=(1+x)^n (1+x^2 )^n & = (1+ ^n C₁ x+ ^n C₂ x^2+ + ^n C_n x^n ) (1+ ^n C₁ x^2+ ^n C₂ x^4+ + ^n C_n x^ 2 n ) aligned The coefficient of x^4 aligned & = ^n C₂+ ^n C₂ ^n C₁+ ^n C₄ & = ^n C₄+ ^n C₂+ ^n C₁ ^n C₂ aligned

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