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

The mean of the coefficients of $x, x^2, \ldots, x^7$ in the binomial expansion of $(2+x)^9$ is $...........$.

The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3) \ldots (1+x^{100})$ is

The coefficient of $x^{50}$ in the binomial expansion of $(1 + x)^{1000} + x(1 + x)^{999} + x^{2}(1 + x)^{998} + \dots + x^{1000}$ is

If $C_{j}$ stands for ${ }^{n} C_{j}$,then $\frac{C_0}{2} + \frac{C_1}{2 \cdot 2^2} + \frac{C_2}{3 \cdot 2^3} + \ldots + \frac{C_{n}}{(n+1) 2^{n+1}} = $

Let $x = (8 \sqrt{3} + 13)^{13}$ and $y = (7 \sqrt{2} + 9)^9$. If $[t]$ denotes the greatest integer $\leq t$,then: