If $(1 + ax)^n = 1 + 8x + 24x^2 + ....,$ then the value of $a$ and $n$ is

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

If $1 + x^4 + x^5 = \sum\limits_{i = 0}^5 a_i (1 + x)^i$ for all $x$ in $\mathbb{R},$ then $a_2$ is

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

For $|x| < 1$,the constant term in the expansion of $\frac{1}{(x-1)^2(x-2)}$ is

$\binom{n}{0} + 2\binom{n}{1} + 2^2\binom{n}{2} + \dots + 2^n\binom{n}{n}$ is equal to