General Term, Middle Term and Greatest Term in Binomial Expansion

Find the term independent of $x$ in the expansion of ${\left[\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{2x^2}\right]}^{10}$

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

For $\beta \ne 0,$ if the coefficient of $x^3$ in the binomial expansion of $(1+\beta x{)}^6$ and the coefficient of $x^4$ in the binomial expansion of $(1-\beta x{)}^8$ are equal, then the value of $\beta$ is

Let $T_r$ denote the $r^{\text{th}}$ term in the binomial expansion of ${\left(a+1\right)}^{50}$. If $T_{25}+T_{27}=\left(\frac{125}{52}\right)T_{26}$, then the sum of all the values of $a$ is

The greatest binomial coefficient in the expansion of ${\left(1+x\right)}^{2n+2}$ $\left(n\in N\right)$ is

Middle term in the expansion of ${\left(x^2+\frac{1}{x^2}+2\right)}^n$ is $(n\in N)$

If the fourth term in the binomial expansion of ${\left(\sqrt{x^{\frac{1}{1+{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}x}}}+x^{\frac{1}{12}}\right)}^6$ is equal to $200,$ and $x>1,$ then the value of $x$ is

In the expansion of ${\left(\frac{a}{x}+bx\right)}^{12},$ the coefficient of $x^{-10}$ will be

The middle term in the expansion of ${\left(x+\frac{1}{2x}\right)}^{2n}$ is

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

The ${10}^{\text{th}}$ term in the expansion of $(x-1{)}^{11}$ (in decreasing powers of $x$) is

The coefficient of $x^{18}$ in the product $\left(1+x\right){\left(1-x\right)}^{10}{\left(1+x+x^2\right)}^9$ is

If $A$ and $B$ are coefficients of $x^{11}$ in the expansion of ${\left(1+x\right)}^{22}$ and ${\left(1+x\right)}^{21}$ respectively, then the value of $\frac{A}{B}$ is

If the term free from $x$ in the expansion of ${\left(\sqrt{x}-\frac{k}{x^2}\right)}^{10}$ is 405, then the value of $k$ is

$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }{x}^{32} \mathrm{i}\mathrm{n}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }{\left(x^4-\frac{1}{x^3}\right)}^{15} \mathrm{i}\mathrm{s}$

 The value of $(\sqrt{5}+2{)}^4+(\sqrt{5}-2{)}^4$ is

If $\alpha$ and $\beta$, be the coefficients of $x^4$ and $x^2$, respectively in the expansion of ${\left(x+\sqrt{x^2-1}\right)}^6+{\left(x-\sqrt{x^2-1}\right)}^6$, then.

The term independent of $x$ in the expansion of ${\left(\frac{2\sqrt{x}}{5}-\frac{1}{2x\sqrt{x}}\right)}^{11}$is

In the expansion of ${\left(1+x\right)}^{29}$ coefficient of ${\left(r+1\right)}^{th}$ term is equal to that of ${\left(r+k\right)}^{th}$ term, then the value of $k$ cannot be 

If ${21}^{st}$ and ${22}^{nd}$ terms in the expansion of ${\left(1+x\right)}^{44}$ are equal, then $x$ is equal to