Given function is $f(y)=1-(y-1)+(y-1)^{2}-(y-1)^{3}$

$+\ldots \ldots-(y-1)^{17}$

In the expansion of $(y-1)^{n}$ 

$T_{r+1}=^{n} C_{r} y^{n-r}(-1)^{r}$

coeffof $y^{2}$ in $(y-1)^{2}=^{2} \mathrm{C}_{0}$

coeff of $y^{2}$ in $(y-1)^{3}=-3^{3} \mathrm{C}_{1}$

coeffof $y^{2}$ in $(y-1)^{4}=^{4} \mathrm{C}_{2}$

So, coeff of termwise is

$^{2} \mathrm{C}_{0}+^{3} \mathrm{C}_{1}+^{4} \mathrm{C}_{2}+^{5} \mathrm{C}_{3}+\ldots \ldots \ldots+^{17} \mathrm{C}_{15}$

$=1+^{3} C_{1}+^{4} C_{2}+^{5} C_{3}+\ldots \ldots \ldots+^{17} C_{15}$

$=\left(^{3} \mathrm{C}_{0}+^{3} \mathrm{C}_{1}\right)+^{4} \mathrm{C}_{2}+^{5} \mathrm{C}_{3}+\ldots \ldots \ldots+^{17} \mathrm{C}_{15}$

$=^{4} \mathrm{C}_{1}+^{4} \mathrm{C}_{2}+^{5} \mathrm{C}_{3}+\ldots \ldots \ldots+^{17} \mathrm{C}_{15}$

$=^{5} \mathrm{C}_{2}+^{5} \mathrm{C}_{3}+\ldots \ldots \ldots+7^{17} \mathrm{C}_{15}$

$=^{18} \mathrm{C}_{15}=18 \mathrm{C}_{3}$