The remainder when $19^{200} + 23^{200}$ is divided by $49$ is $.........$.

$x^5 + 10x^4a + 40x^3a^2 + 80x^2a^3 + 80xa^4 + 32a^5 = $

Let $(a + bx + cx^2)^{10} = \sum_{i=0}^{20} p_i x^i$,where $a, b, c \in N$. If $p_1 = 20$ and $p_2 = 210$,then $2(a + b + c)$ is equal to

If $b$ is very small as compared to the value of $a$,so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+\ldots+\frac{1}{a-nb}=\alpha n+\beta n^2+\gamma n^3$,then the value of $\gamma$ 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 $a+b+c+d=$

The coefficient of $x^{50}$ in the expansion of $(1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \ldots + x^{1000}$ is