The value of $\sum\limits_{r = 1}^{19} {\frac{{{}^{20}{C_{r + 1}}{(-1)}^r}}{{{2^{2r + 1}}}}}$ is

Let $a$ be the largest real root and $b$ be the smallest real root of the polynomial equation $x^6-6x^5+15x^4-20x^3+15x^2-6x+1=0$. Then $\frac{a^2+b^2}{a+b+1}$ is

The value of $(\sqrt{2} + 1)^6 + (\sqrt{2} - 1)^6$ is

Assertion $(A)$: If $a_1, a_2, \ldots, a_n$ are the $n$ distinct roots of the equation $x^n-2=0$,then $1+\left(1-a_1\right)\left(1-a_2\right) \ldots\left(1-a_n\right)=0$.
Reason $(R)$: If $\alpha_1, \alpha_2, \ldots, \alpha_n$ are the roots of $f(x) \equiv p_0 x^n+p_1 x^{n-1}+\ldots+p_n=0$,then the roots of $f(g(x))=0$ are $g^{-1}(\alpha_i)$ for $i=1, 2, \ldots, n$.
The correct option among the following is:

The coefficient of $x^{50}$ in the binomial expansion of $(1 + x)^{1000} + x(1 + x)^{999} + x^{2}(1 + x)^{998} + \dots + x^{1000}$ is

Compute $(98)^{5}$.