11. Let the polynomial $1-x+x^{2}-x^{8}+\cdots+x^{16}$ $-x^{17}$ be written as $a_{0}+a_{1} y+a_{2} y^{2}+a_{3} y^{3}+\cdots$ $+a_{10} y^{16}+a_{17} y^{17}$,

where $y=x+1$, and all $a_{\mathrm{i}}$ are constants, find $a_{2}$.
$$
\begin{array}{l}
\text { 11. Replace } x \text { with } y-1 \text {, the polynomial becomes } \\
\text { 1- }(y-1)+(y-1)^{2}-(y-1)^{3} \\
+\cdots+(y-1)^{10}-(y-1)^{17}, \\
\quad 1+(1-y)+(1-y)^{2}+(1-y)^{3} \\
+\cdots+(1-y)^{10}+(1-y)^{17} .
\end{array}
$$

Notice that the coefficient of $y^{2}$ in $(1-y)^{\mathrm{k}}$ is $C_{\mathrm{k}}^{2}$ $(2 \leqslant k \leqslant 17)$, so the required number is
$$
\begin{array}{l}
C_{2}^{2}+C_{3}^{2}+C_{4}^{2}+\cdots+C_{17}^{2} \\
=C_{18}^{3}=816 .
\end{array}
$$