The number of terms in the expansion of $(x^{2}+y^{2})^{25}-(x^{2}-y^{2})^{25}$ after simplification is

The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3) \ldots (1+x^{100})$ is

What is the sum of the coefficients of $(x^2 - x - 1)^{99}$?

If $a_k$ is the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n$ for $k=0, 1, 2, \ldots, 2n$,then $a_1+2a_2+3a_3+\ldots+2na_{2n}$ is equal to

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 sum of the coefficients of the terms of degree $m$ in the expansion of $(1 + x)^n(1 + y)^n(1 + z)^n$ is: