General Term in Binomial Expansion

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

Using binomial expansion, prove that ${\left(101\right)}^{50}>{100}^{50}+{99}^{50}$.

Which number is larger ${\left(1.2\right)}^{4000}$ or $800$?

Which is larger ${99}^{50}+{100}^{50}$ or ${\left(101\right)}^{50}$?

Find an approximation of ${\left(0.99\right)}^5$ using the first three terms of its expansions. [Write the answer up to three decimal places.]

Which is larger ${\left(1.01\right)}^{1000000}$ or $10,000$?

The coefficient of $x$ in the expansion of $\left(1-3x+7x^2\right){\left(1-x\right)}^{16}$ is

Find the ratio between the constant term and the coefficient of the fifth term in the expansion of ${\left(\frac{1}{x}+x\right)}^6$.

Find the $8^{th}$ term of ${\left(1-\frac{5x}{2}\right)}^{\frac{-3}{5}}$.

If the coefficient of $x^{11}$ in ${\left(2x^2+\frac{3}{x^3}\right)}^{13}$ is $\left(\begin{array}{c}m\\ n\end{array}\right)\times 2^{10}\times 3^3$, then value of $m+n$ is 

If $n$ is a positive integer then the coefficient of $x^6$ in the expansion of ${\left(1-2x+3x^2-4x^3+\dots \dots \right)}^{-n}$ is

If the coefficients of $(2\alpha +5{)}^{\mathrm{th}}$ and $(\alpha -1{)}^{\text{th}}$ terms in the expansion $(1+x{)}^{2019}$ are equal, then $\alpha =$

The coefficients of $x^{50}$ in ${\left(1+x\right)}^{101}{\left(1-x+x^2\right)}^{100}$ is

Find the term independent of $x$ in the expansion of $\left(1+x+2x^2\right){\left(\frac{3x^2}{2}-\frac{1}{3x}\right)}^4$.

Let $n$ be a positive integer. If the coefficients of $2^{nd}$, $3^{rd}$, $4^{th}$ terms in the expansion of $(1+x{)}^n$ are in $\mathrm{A}.\mathrm{P}.$, then find the value of $n$.

If the coefficients of $4^{th}$ and ${13}^{th}$ terms in the expansion of ${\left(x^2+\frac{1}{x}\right)}^n$ are equal, then find the term which is independent of $x$.

In the binomial expansion of $(1+x{)}^{10}$, the coefficients of $(2m+1{)}^{th}$ and $(4m+5{)}^{th}$ are equal. Find $m$.

In the expansion of ${\left(x^2-\frac{1}{3x}\right)}^9$, find the independent of $x$.

In the expansion of ${\left(x^2-\frac{1}{3x}\right)}^9$, find the coefficient of $x^9$.

The term independent of $x$ in the expansion of ${(1-x)}^2{(x+\frac{1}{x})}^{10}\mathrm{}$ is