General Term, Middle Term and Greatest Term in Binomial Expansion

If the coefficients of ${\left(2r+4\right)}^{\mathrm{th}},{\left(r-2\right)}^{\mathrm{th}}$ terms in the expansion of ${\left(1+x\right)}^{18}$ are equal, then the value of $r$ is

The value of the numerically greatest term in the expansion of $(2x+3y{)}^{11}$ when $x=\frac{1}{2}$ and $y=\frac{1}{3}$ is

The number of irrational terms in the expansion of ${\left(5^{\frac{1}{6}}+2^{\frac{1}{8}}\right)}^{100}$ is

The sum of all those terms which are rational numbers in the expansion of ${\left(2^{\frac{1}{3}}+3^{\frac{1}{4}}\right)}^{12}$ is:

For the natural numbers $m, n,$ if $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+\dots .+a_{m+n}{y}^{m+n}$ and $a_1=a_2=10,$ then the value of $m, n,$ is equal to:

If the term independent of $x$ in the expansion of ${\left[\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right]}^{10}, x\ne 1$, is equal to $k^{th}$ term, then the value of $k$ is 

In the expansion of ${\left(\sqrt{x}+\frac{1}{3x^2}\right)}^{10}$ the value of constant term (independent of $x$ ) is

The value of the term independent of $x$ in the expansion of ${\left(x^2-\frac{1}{x}\right)}^9$ is

If the independent term in the expansion of ${\left(\sqrt{x}+\frac{k}{x^2}\right)}^{10}$ is $405,$ then consider the following statements.
$\mathrm{I}.$ The third term is independent of $x$.
$\mathrm{II}.$ The value of $k$ is $\pm 3.$ .
$\mathrm{III}.$ Total number of terms in the expansion is $10.$

Which of the above statement(s) is/are correct?

If the coefficient of the middle term in the expansion of $(1+x{)}^{2n+2}$ is $p$ and the coefficients of middle terms in the expansion of $(1+x{)}^{2n+1}$ are $q$ and $r,$ then

What is the term independent of $x$ in the expansion of $\left(1+x+2x^2\right){\left(\frac{3x^{-2}}{2}-\frac{1}{3x}\right)}^9$?

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

What is the coefficient of $x^4$ in the expansion of ${\left(\frac{1-x}{1+x}\right)}^2?$

The coefficient of $x^{-n}$ in ${\left(1+x\right)}^n{\left(1+\frac{1}{x}\right)}^n$ is

The coefficient of $x^4$ in the expansion of ${\left(1+x+x^2+x^3\right)}^{11}$ is

The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+\dots +(1+x{)}^{30}$ is

The coefficient of $x^{53}$ in the following expansion $\sum _{m=0}^{100}{}^{100}C_m{\left(x-3\right)}^{100-m}·2^m$ is

The expression ${\left(\sqrt{2x^2+1}+\sqrt{2x^2-1}\right)}^6+{\left(\frac{2}{\sqrt{2x^2+1}+\sqrt{2x^2-1}}\right)}^6$ is a polynomial of degree

The coefficient of the middle term in the binomial expansion in powers of $x$ of ${\left(1+\alpha x\right)}^4$ and of ${\left(1-\alpha x\right)}^6$ is the same, if $\alpha$ equals

If the coefficients of $r^{\mathrm{th}}, {\left(r+1\right)}^{\mathrm{th}}$ and ${\left(r+2\right)}^{\mathrm{th}}$ terms in the binomial expansion of ${\left(1+y\right)}^m$ are in $\mathrm{AP},$ then $m$ and $r$ satisfy the equation