The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + \alpha x)^4$ and of $(1 - \alpha x)^6$ is the same if  $\alpha$ equals

If the coefficients of $x^{7}$ in $\left(x^{2}+\frac{1}{b x}\right)^{11}$ and $x^{-7}$ in $\left(x-\frac{1}{b x^{2}}\right)^{11}, b \neq 0$, are equal, then the value of $b$ is equal to:

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is :

In the expansion of the following expression $1 + (1 + x) + {(1 + x)^2} + ..... + {(1 + x)^n}$ the coefficient of ${x^k}(0 \le k \le n)$ is

The coefficient of the term independent of $x$ in the expansion of ${\left( {\sqrt {\frac{x}{3}}  + \frac{3}{{2{x^2}}}} \right)^{10}}$ is

The coefficient of the term independent of $x$ in the expansion of $(1 + x + 2{x^3}){\left( {\frac{3}{2}{x^2} - \frac{1}{{3x}}} \right)^9}$ is