General and Middle Terms in Binomial Expansion

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

The sum of the coefficients of three consecutive terms in the binomial expansion of ${\left(1+x\right)}^{n+2}$, which are in the ratio $1:3:5$, is equal to

The coefficient of $x^{18}$ in the expansion of ${\left(x^4-\frac{1}{x^3}\right)}^{15}$ is $\_\_\_\_\_\_\_\_\_\_\_\_$

Coefficient of $x^4$ in ${\left(2x^3-\frac{1}{3x^8}\right)}^5$ is

If the ratio of three consecutive terms in the binomial expansion of ${\left(1+x\right)}^{n+2}$ is $1:3:5$, then sum of consecutive terms is

If the coefficient of $x^7$ in the expansion of ${\left(ax-\frac{1}{b{x}^2}\right)}^{13}$ is equal to the coefficient of $x^{-5}$ in the expansion of ${\left(ax+\frac{1}{b{x}^2}\right)}^{13}$, then $a^4{b}^4$ is

The ratio of $5^{\mathrm{th}}$ term from the beginning and $5^{\mathrm{th}}$ term from the end is $\sqrt{6}:1$ in ${\left(2^{\frac{1}{4}}+3^{-\frac{1}{4}}\right)}^n$, then the value of $n$ is

${}^{30}C_0{}^{20}C_{10}+{}^{31}C_1{}^{19}C_{10}+{}^{32}C_2{}^{18}C_{10}+....+{}^{40}C_{10}{}^{10}C_{10}$ is equal to

The expression ${\left[x+{\left(x^3-1\right)}^{\frac{1}{2}}\right]}^5+{\left[x-{\left(x^3-1\right)}^{\frac{1}{2}}\right]}^5$ is a polynomial of degree

If three dices are thrown together, then the number of ways in which sum of the numbers appearing on the dice is $n$, $9\le n\le 14$ is

If the coefficient of $r^{\mathrm{th}}$ term and ${\left(r+1\right)}^{\mathrm{th}}$ term in the binomial expansion of ${\left(1+x\right)}^{20}$ are in the ratio $1:2$, then $r$ is equal to

If $n$ is even and ${}^{n}C_0<{}^{n}C_1<{}^{n}C_2<....<{}^{n}C_r>{}^{n}C_{r+1}>{}^{n}C_{r+2}>...>{}^{n}C_n$, then $r=$

The term independent of $x$ in the expansion of ${\left(1+x+x^{-2}+x^{-3}\right)}^{10}$ is

If ${\left(\sqrt{3}+i\right)}^8-{\left(\sqrt{3}-i\right)}^8=\alpha +i\beta$, then $\alpha -\frac{\sqrt{3}}{2}\beta$ 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

The coefficient of $x^{17}$ in the expansion of $\left(x-1\right)\left(x-2\right)\left(x-3\right)...(x-18)$ is

The number of irrational terms in expansion of ${\left(4^{\frac{1}{5}}+7^{\frac{1}{10}}\right)}^{45}$ is

If, in the expansion of ${\left(1+x\right)}^{20}$, the co-efficients of $\left(r+1\right)$th and $\left(r+3\right)$th terms are equal, then the value of $r$ is

The sum of the co-efficients of the $r$th and $\left(r+1\right)$th terms in the expansion of ${\left(1+x\right)}^n$ is, in the expansion of ${\left(1+x\right)}^{n+1}$, the co-efficient of