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.

${}^{12}C_8$

Which of the following will be true ?

The coefficient of $y^{10}$ in the polynomial $\left(y+{}^{10}C_0\right) \left(y+3·{}^{10}C_1\right) \left(y+5·{}^{10}C_2\right)......\left(y+21·{}^{10}C_{10}\right)$ is

Coefficient of $x^8$ in the expansion of ${\left(1+x\right)}^4{\left(1+x^2\right)}^5{\left(1+x^4\right)}^{10}{\left(1-x\right)}^4$ is

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

Which of the following will be true ?

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

Let $M=2^{30}-2^{15}+1,$ and $M^2$ be expressed in base $2.$ The number of $1^{\text{'}}\text{s}$ in this base $2$ representation of $M^2$ is

Let $r$ be the remainder when ${2021}^{2020}$ is divided by ${2020}^2$. Then $r$ lies between

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 $n$.