# The coefficient of $x^{17}$ in the expansion of $(x-1)(x-2)\times (x-3)$.......$(x-18)$ is

$\begin{array}{1 1}(A)\;342\\(B)\;-171\\(C)\;\large\frac{171}{2}\\(D)\;684\end{array}$

Toolbox:
• $T_{r+1}=nC_r a^{n-r} b^r$
Coefficient of $x^{17}$ in $(x-1)(x-2)(x-3)......(x-18)$
$\Rightarrow -1-2-3+........+(-18)$
$\Rightarrow -(1+2+3+......+18)$
$\Rightarrow -\large\frac{18\times (18+1)}{2}$
$\Rightarrow -171$
Hence (B) is the correct answer.