Find the coefficient of $a^{4}$ in the product $(1+2a)^{4}(2-a)^{5}$ using the binomial theorem.

Let the smallest value of $k \in N$,for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + \dots + (1+x)^{99} + (1+kx)^{100}, x \neq 0$,is $(43n + \frac{101}{4}) ({}^{100}C_3)$ for some $n \in N$,be $p$. Then the value of $p+n$ is:

Let $K$ be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[3]{3})^{6144}$. If the coefficient of $x^{P} \quad(P \in N)$ in the expansion of $\frac{1}{(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})}$ is $\alpha_{P}$,then $\alpha_{K}-\alpha_{K+1}-\alpha_{K-1}=$

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:

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+.......+x^{1000}$ is

If ${T_0}, {T_1}, {T_2}, \dots, {T_n}$ represent the terms in the expansion of ${(x + a)^n}$,then $({T_0} - {T_2} + {T_4} - \dots)^2 + ({T_1} - {T_3} + {T_5} - \dots)^2 = $