If $(1+x+x^2)^n = c_0 + c_1 x + c_2 x^2 + \ldots$,then the value of $c_0 c_1 - c_1 c_2 + c_2 c_3 - \ldots$ is

Let $R=(5 \sqrt{5}+11)^{2 n+1}$ and $f=R-[R]$,where $[x]$ denotes the greatest integer less than or equal to $x$,then $R f=$

If the $6^{th}$ term in the expansion of the binomial $[\sqrt{2^{\log(10 - 3^x)}} + \sqrt[5]{2^{(x - 2)\log 3}}]^m$ is equal to $21$ and it is known that the binomial coefficients of the $2^{nd}$,$3^{rd}$ and $4^{th}$ terms in the expansion represent respectively the first,third and fifth terms of an $A.P.$ (the symbol $\log$ stands for logarithm to the base $10$),then $x = $

If the coefficient of $x$ in the expansion of $(ax^{2}+bx+c)(1-2x)^{26}$ is $-56$ and the coefficients of $x^{2}$ and $x^{3}$ are both zero, then $a+b+c$ is equal to:

The coefficient of $x^{13}$ in the expansion of $(1 - x)^5(1 + x + x^2 + x^3)^4$ is :-

The correct matching of List-$I$ from List-$II$ is: