The coefficient of x^{18} in the product \left ( 1+x \right )\left ( 1-x \right )^{10}\left ( 1+x+x^{2} \right )^{9} is : 

  • Option 1)

    -84

  • Option 2)

    84

  • Option 3)

    126

  • Option 4)

    -126

Answers (1)

\left ( 1+x \right )\left ( 1-x \right )^{10}\left ( 1+x+x^{2} \right )^{9}

=\left ( 1+x \right )\left ( 1-x \right )\left ( 1-x \right )^{9}\left ( 1+x=x^{2} \right )^{9}

=\left ( 1-x^{2} \right )\left ( 1-x^{3} \right )^{9}

So coefficient of   x^{18}=_{}^{9}\textrm{C}_{6}=84

 

 


Option 1)

-84

Option 2)

84

Option 3)

126

Option 4)

-126

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