Let $(1 - 2x + 3x^2)^{10} = a_0 + a_1x + a_2x^2 + \dots + a_n x^n$,where $a_n \neq 0$. Then the arithmetic mean of $a_0, a_1, a_2, \dots, a_n$ is

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x+\sqrt{x^3-1})^5+(x-\sqrt{x^3-1})^5, x>1$. If $u$ and $v$ satisfy the equations $\alpha u+\beta v=18$ and $\gamma u+\delta v=20$,then $u+v$ equals:

The coefficient of $x^4$ in the expansion of $(1+x-x^2-x^3)^{11}$ is

$(102)^4 = ?$

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

When $x$ is so small that its square and its higher powers may be neglected,then the value of $\frac{\left(1+\frac{3}{4} x\right)^{-4} \sqrt{(3+x)}}{\sqrt{(3-x)^3}}$ is approximately equal to