Question

If $f\left(x\right)=1-x+x^2-x^3+\cdots -x^{15}+x^{16}-x^{17},$ then the coefficient of $x^2$ in $f(x-1)$ is

Answer 1

$f\left(x\right)=1-x+x^2-x^3+\cdots -x^{15}+x^{16}-x^{17}$
$\Rightarrow f\left(x\right)=\frac{1-x^{18}}{1+x}$
$\Rightarrow f(x-1)=\frac{1-(x-1{)}^{18}}{1+(x-1)}$
Therefore, required coefficient of $x^2$ is equal to the coefficient of $x^3$ in $1-(1-x{)}^{18},$ which is given by ${}^{18}{C}_3=816$