The coefficient of $x^n$ in the expansion of $(1 + x + x^2 + ....)^{-n}$ is

If $a_k$ is the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n$ for $k=0, 1, 2, \ldots, 2n$,then $a_1+2a_2+3a_3+\ldots+2na_{2n}$ is equal to

The coefficient of $x^8$ in the expansion of $(x-1)(x-2)(x-3)\ldots(x-10)$ is:

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

The sum of the coefficients of the terms of degree $m$ in the expansion of $(1 + x)^n(1 + y)^n(1 + z)^n$ is:

If the roots of $x^5-ax^4+bx^3-cx^2+dx-1=0$ are all positive such that their arithmetic mean and geometric mean are equal,then $a+b+c+d=$