General and Middle Terms

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

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{15}C_5$

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{20}C_7$

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{10}C_9$

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{8}C_6$

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{12}C_8$

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

The constant term in the expansion of ${\left[x^2-\left(\frac{1}{x^2}\right)\right]}^{16}$ is

In the expansion of ${\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)}^n$, the ratio of $7^{th}$ term from the beginning to the $7^{th}$ term from the end is $1: 6$, find $n$.

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 $(1+x{)}^{10}$, if the coefficients of $(2m+1{)}^{th}$ and $(4m+5{)}^{th}$ terms are equal then, find $m$.

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

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

If three consecutive coefficients in the expansion of $(1+x{)}^n$ are in the ratio $6: 33: 110$, find $n$.

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 

Let $n\in N$ which one of the following is true?

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 $n$.