The coefficient of $x^2$ in the expansion of the product $(2 - x^2)((1 + 2x + 3x^2)^6 + (1 - 4x^2)^6)$ is

If $a$ and $b$ are distinct integers,prove that $a-b$ is a factor of $a^{n}-b^{n}$,whenever $n$ is a positive integer.

If $f(x)=(2011+x)^n$,where $x$ is a real variable and $n$ is a positive integer,then the value of $f(0)+f^{\prime}(0)+\frac{f^{\prime \prime}(0)}{2 !}+\ldots+\frac{f^{(n-1)}(0)}{(n-1) !}$ is

Let $n > 2$ be an integer and define a polynomial $p(x) = x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$,where $a_0, a_1, \ldots, a_{n-1}$ are integers. Suppose we know that $n p(x) = (1 + x) p'(x)$. If $b = p(1)$,then:

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

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