For $x \in R, x \neq -1,$ if $(1 + x)^{2016} + x(1 + x)^{2015} + x^2(1 + x)^{2014} + \dots + x^{2016} = \sum_{i = 0}^{2016} a_i x^i,$ then $a_{17}$ is equal to

$(1 + x)^n - nx - 1$ is divisible by (where $n \in N$):

Let $S=\{a+b \sqrt{2}: a, b \in Z \}$,$T_1=\{(-1+\sqrt{2})^n: n \in N \}$ and $T_2=\{(1+\sqrt{2})^n: n \in N \}$. Then which of the following statements is (are) $TRUE$?
$(A)$ $Z \cup T_1 \cup T_2 \subset S$
$(B)$ $T_1 \cap (0, \frac{1}{2024}) = \phi$,where $\phi$ denotes the empty set
$(C)$ $T_2 \cap (2024, \infty) \neq \phi$
$(D)$ For any given $a, b \in Z$,$\cos(\pi(a+b \sqrt{2})) + i \sin(\pi(a+b \sqrt{2})) \in Z$ if and only if $b=0$,where $i=\sqrt{-1}$

The greatest integer less than or equal to $(\sqrt{3}+2)^5$ is

The coefficient of $x^{50}$ in $(1+x)^{101} (1-x+x^2)^{100}$ is......

If $(1+x+x^2+x^3)^5 = \sum_{k=0}^{15} a_k x^k$,then $\sum_{k=0}^7 a_{2k}$ is equal to