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

If $A$ denotes the sum of all the coefficients in the expansion of $(1-3x+10x^2)^n$ and $B$ denotes the sum of all the coefficients in the expansion of $(1+x^2)^n$,then:

$x^5 + 10x^4a + 40x^3a^2 + 80x^2a^3 + 80xa^4 + 32a^5 = $

If the number of terms in the expansion of $(x - 2y + 3z)^n$ is $45$,then $n=$

If the ratio of the coefficients of the third and fourth terms in the expansion of $(x - \frac{1}{2x})^n$ is $1 : 2$,then the value of $n$ 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: