The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+.......+x^{1000}$ is

If $(1+x+x^2+x^3)^5 = \sum_{k=0}^{15} a_k x^k$, then $\sum_{k=0}^7 (-1)^k a_{2k}$ is equal to

If the coefficients of $x$ and $x^{2}$ in the expansion of $(1+x)^{p}(1-x)^{q}$,where $p, q \leq 15$,are $-3$ and $-5$ respectively,then the coefficient of $x^{3}$ is equal to $............$

If the coefficient of $x$ in the expansion of $(ax^{2}+bx+c)(1-2x)^{26}$ is $-56$ and the coefficients of $x^{2}$ and $x^{3}$ are both zero, then $a+b+c$ is equal to:

The coefficient of $x^{2012}$ in the expansion of $(1-x)^{2008}(1+x+x^2)^{2007}$ is equal to

Let the smallest value of $k \in N$, for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + \dots + (1+x)^{99} + (1+kx)^{100}, x \neq 0$, is $(43n + \frac{101}{4}) ({}^{100}C_3)$ for some $n \in N$, be $p$. Then the value of $p+n$ is: