The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^{2}+x+1\right)^{100}$ is:

Given $(1 - 2x + 5x^2 - 10x^3) (1 + x)^n = 1 + a_1x + a_2x^2 + ....$ and that $a_1^2\,= 2a_2$ then the value of $n$ is

If ${S_n} = \sum\limits_{r = 0}^n {\frac{1}{{^n{C_r}}}} $ and ${t_n} = \sum\limits_{r = 0}^n {\frac{r}{{^n{C_r}}}} $, then $\frac{{{t_n}}}{{{S_n}}}$ is equal to

The coefficient of $x^{4}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{6}$ in powers of $x,$ is

The sum of coefficients in the expansion of ${(1 + x + {x^2})^n}$ is

In the polynomial $(x - 1)(x - 2)(x - 3).............(x - 100),$ the coefficient of ${x^{99}}$ is