Question

find the coefficient of $x^{7}$ in $(1+x+x^2+..+x^7+x^9-x)^{11}$
correct answer with proper and full solution please
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Answer 1

$1+x+x^2+..+x^7+x^9-x=1+x^2+x^3+,,x^7+x^9 \\ =1+x^2+x^3(1+x^2)+x^4(1+x^2)+x^7(1+x^2) \\ =(1+x^2)(1+x^3+x^4+x^7)=(1+x^2)(1+x^3)(1+x^4)$
$((1+x^2)(1+x^3)(1+x^4))^{11}=(1+x^2)^{11}(1+x^3)^{11}(1+x^4)^{11}$
So for coefficent of x⁷ 
     x^4 from  (1+x⁴)¹¹ and x³ from (1+x³)¹¹
    x⁴ from (1+x²)¹¹ and x³ from (1+x³)¹¹   
  $=C^{11}_{1}C^{11}_{1}+C^{11}_{2}C^{11}_{1}=11(11)+(11*5)11=726$

Answer 2

   

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