Binomial Theorem

Let $\left[x\right]$ denote greatest integer less than or equal to $x$. If for $n\in\,N,\left(1-x+x^{3}\right)^{n}=\underset{j=0}{\overset{3n}{\sum\,}}a_{j}x^{j}$,then…

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

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

If the fourth term in the Binomial expansion of $\left(\frac{2}{x}+x^{log_{8}x}\right)^{6},\left(x>0\right)$ is $20\times\,8^{7},$then a value of $x$ is

For a positive integer$n,\left(1+\frac{1}{x}\right)^{n}$ is expanded in increasing powers of $x$. If three consecutivecoefficients in this expansion are in the ratio, $2:5:12,$ then $n$ is equal to

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:

In the expansion of $\left(\frac{x}{\cos\,\theta\,}+\frac{1}{x\sin\,\theta\,}\right)^{16},$ if $l_{1}$ is the least value of the term independent of $x$ when $\frac{\pi\,}{8}\leq\,\theta\,\leq\,\frac{\pi\,}{4}$ and…

The mean of the coefficients of $x,x^{2},\ldots\,\ldots\,,x^{7}$in the binomial expression of $(2+x)^{9}$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…

If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of $\left(x^{n}+\frac{2}{x^{5}}\right)^{7}$is $939$, then the sum of all the possible integral values of$n$ is

The sum of coefficients of integral powers of $x$in the binomial expansion of $\left(1-2\sqrt{x}\right)^{\text{50}}$is

The coefficient of $x^{101}$in the expression $\left(5+x\right)^{500}+x\left(5+x\right)^{499}+x^{2}\left(5+x\right)^{498}+\ldots\,\ldots\,+x^{500},x>0$ is