If the coefficient of $x^r$ in the expansion of $(1+x+x^2+x^3)^{100}$ is $a_r$,and $S = \sum_{r=0}^{300} a_r$,then $\sum_{r=0}^{300} r \cdot a_r =$

If the coefficients of $x$ and $x^{2}$ in the expansion of $(1+x)^{p}(1-x)^{q}$,where $p, q \leq 15$,are $-3$ and $-5$ respectively,then the coefficient of $x^{3}$ is equal to $............$

Let $(5 + 2\sqrt{6})^n = p + f$,where $n \in N$,$p \in N$,and $0 < f < 1$. Then the value of $f^2 - f + pf - p$ is:

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

The fractional part of a real number $x$ is defined as $x - [x]$,where $[x]$ is the greatest integer less than or equal to $x$. Let $F_1$ and $F_2$ be the fractional parts of $(44 - \sqrt{2017})^{2017}$ and $(44 + \sqrt{2017})^{2017}$,respectively. Then,$F_1 + F_2$ lies between the numbers:

Let $(7 + 4\sqrt{3})^n = p + \beta$,where $n$ and $p$ are positive integers and $\beta \in (0, 1)$. Then $(1 - \beta)(p + \beta)$ is