If $n$ is a positive integer,then the coefficient of $x^6$ in the expansion of $(1-2x+3x^2-4x^3+\ldots)^{-n}$ is

The sum of the coefficients in the expansion of $(1 + x - 3x^2)^{2163}$ will be

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 last three digits of the number $N = 7^{100} - 3^{100}$ are

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

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}} = $