5-12 Let the polynomial $1-x+x^{2}-x^{3}+\cdots+x^{16}-x^{17}$ be written as $a_{0}+a_{1} y+a_{2} y^{2}+a_{3} y^{3}+\cdots+a_{16} y^{16}+a_{17} y^{17}$, where $y=x+1$, and each $a_{i}$ is a constant, find $a_{2}$.
[Solution]
$$
\begin{aligned}
& 1-x+x^{2}-x^{3}+\cdots+x^{16}-x^{17} \\
= & \frac{1-x^{18}}{1+x} \\
= & \frac{1-(y-1)^{18}}{y} .
\end{aligned}
$$

In the numerator of the above expression, the coefficient of $y^{3}$ is $C_{18}^{3}$, so
$$
a_{2}=C_{18}^{3}=816 .
$$