Binomial Theorem

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:

Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^{2}}\right)^{n},x\neq\,0,$ be in the ratio $12:8:3.$Then the term independent of $x$ in the expansion, is equal to _______.

The total number of irrational terms in the binomial expansion of $\left(7^{\frac{1}{5}}-3^{\frac{1}{10}}\right)^{60}$ is

If the constant term in the binomial expansion of$\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}$is$405$, then$\left|k\right|$equals :

If $^{n}C_{4},^{n}C_{5}$and $^{n}C_{6}$ are in A.P.,then $n$ can be

If the co-efficient of $x^{9}$in $\left(\alpha\,x^{3}+\frac{1}{\beta\,x}\right)^{11}$and theco-efficient of $x^{-9}$ in $\left(\alpha\,x-\frac{1}{\beta\,x^{3}}\right)^{11}$ are equal, then$(\alpha\,\beta\,)^{2}$is equal…

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2x)^{n}$ be in the ratio $2:5:8$. Then the coefficient of the term, which is in the middle of these three terms, is

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{l}}\right)^{9}$ is $-84$ and the coefficient of $x^{-3l}$ is $2^{\alpha\,}\beta\,$ where $\beta\,<0$ is an odd number, then…

A ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of $\left(2^{\frac{1}{3}}+\frac{1}{2\left(3\right)^{\frac{1}{3}}}\right)^{10}$ is

Let $x=\left(8\sqrt{3}+13\right)^{13}$and $y=\left(7\sqrt{2}+9\right)^{9}$. If $\left[t\right]$ denotes the greatest integer $\leq\,t$, then

If the coefficients of $x^{7}$ in$\left(x^{2}+\frac{1}{bx}\right)^{11}$and $x^{-7}$in $\left(x-\frac{1}{bx^{2}}\right)^{11},b\neq\,0,$are equal, then the value of $b$is equal to:

If$Prn=Pr+1n$ and$Crn=Cr-1n,$then the value of$r$is equal to: