Binomial Theorem for Positive Integral Index

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

If $L$ and $M$ are respectively the coefficient of $x^{-7}$ in ${\left(ax+\frac{b}{x^2}\right)}^{11}$ and the coefficient of $x^7$ in ${\left(b{x}^2+\frac{a}{x}\right)}^{11}$ then $L+M=$

If the $4^{\mathrm{th}}$ term in the expansion of ${\left(\frac{x}{2}-\frac{2y}{3}\right)}^6$ is $-20$, then $xy=$

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

If the digits at ten's and hundred's places in ${\left(11\right)}^{2016}$ are $x$ and $y$ respectively, then the ordered pair $\left(x,y\right)$ is equal to

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

If the coefficient of$7^{\mathrm{th}}$ and ${13}^{\mathrm{th}}$ terms in the expansion of ${\left(1+x\right)}^n$, $\left(n\in N\right)$ are equal, then $n$ is equal to

In the expansion of ${\left(1+x\right)}^{15}+\mathrm{ }{\left(1+x\right)}^{16}+\mathrm{ }{\left(1+x\right)}^{17}+...+\mathrm{ }{\left(1+x\right)}^{30}$, the coefficient of $x^{10}$ 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

The coefficient of $x^4$ in the expansion of ${\left(\frac{x}{2}-\frac{3}{x^2}\right)}^{10}$ 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

The coefficient of $x^{\mathit{n}}$ in the expansion of ${\left(1+x\right)}^{2n}$ and ${\left(1+x\right)}^{2n-1}$ are in the ratio