For Coefficient (1)

Relevant wiki: Hockey Stick Identity

The coefficient is given by: $\dbinom{20}{18}+\dbinom{19}{17}+\dbinom{18}{16}+\cdots+\dbinom{2}0$ $\large =\displaystyle\sum_{n=2}^{20}\dbinom{20}{n-2}=\displaystyle\sum_{n=2}^{20}\dbinom{20}{2}$

Using hockey stick identity , this is:- $\large\dbinom{21}{3}=\color{#D61F06}{\boxed{1330}}$

This Solution is similar to @Rishabh Cool. I am just simplifying the above expression , which makes this problem Super Easy.

After Simplification i will use Binomial Thm to find the coefficient of $x^{18}$ . We have,

$P(x)=(1+x)^{20}+x(1+x)^{19}+\cdots+x^{17}(1+x)^3+x^{18}(1+x)^2 \\ -\dfrac{x}{1+x}\times P(x)=-x(1+x)^{19}-\cdots-x^{17}(1+x)^3-x^{18}(1+x)^2-x^{19}(1+x)$

Add the Two Equations , we will get

$(1-\dfrac{x}{1+x})P(x)=(1+x)^{20}-x^{19}(1+x) \\ P(x)=(1+x)^{21}-x^{19}(1+x)^2 \\ \text{Coefficient of }x^{18} \text{ in the Above expression is } \dbinom{21}{18}\\ \text{Our Answer : } \boxed{1330}$