Binomial Theorem

If the co-efficient of $x^{7}$ and $x^{8}$ in the expansion of$\left(2+\frac{x}{3}\right)^{n}$ are equal, then the value of $n$ is equal to :

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{n}$, in the increasing powers of…

If$C32n:C3n=10:1,$then the ratio$\left(n^{2}+3n\right):\left(n^{2}-3n+4\right)$is

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

A possible value of$x,$for which the ninth term in the expansion of $\left{3^{\log\,_{3}\sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right)\log\,_{3}\left(5^{x-1}+1\right)}\right}^{10}$ in the increasing powers of…

Suppose$\underset{r=0}{\overset{2023}{\sum\,}}r^{2}\cdot\,Cr2023=2023\times\,\alpha\,\times\,2^{2022}$, then the value of$\alpha\,$is

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

Let $m\text{and}n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of$\left(\frac{1}{3}x^{\frac{1}{3}}+\frac{1}{2x^{\frac{2}{3}}}\right)^{18}$.…

What is $C(n, 1)+C(n, 2)+\ldots+C(n, n)$ equal to?

What is $\sum_{r=0}^n 2^r C(n, r)$ equal to?

What is the sum of all the coefficients in the expansion of $(1+x)^n$ ?