Binomial Theorem

Let $n\in\,N$ and $\left[x\right]$ denote the greatest integer less than or equal to $x.$ If the sum of $\left(n+1\right)$ terms of $C0n,3\cdot\,C1n,5\cdot\,C2n,7\cdot\,C3n,\ldots\,$ is equal to$2^{100}\cdot\,101,$then…

If the sum of the co-efficients of all the positive even powers of$x$ in the binomial expansion of $\left(2x^{3}+\frac{3}{x}\right)^{10}$is $5^{10}-\beta\,\cdot\,3^{9}$, then $\beta\,$ is equal to _____.

The sum of the co-efficient of all odd degree terms in the expansion of $\left(x+\sqrt{x^{3}-1}\right)^{5}+\left(x-\sqrt{x^{3}-1}\right)^{5},\left(x>1\right)$ is

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…

$3\times\,7^{22}+2\times\,10^{22}-44$when divided by $18$ leaves the remainder

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

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

The sum $\underset{r=1}{\overset{10}{\sum\,}}\left(r^{2}+1\right)\times\,\left(r!\right)$,is equal to:

The sum of the co-efficient of all even degree terms in $x$ in the expansion of $\left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6},\left(x>1\right)$ is equal to

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…

If $Cr20$ is the co-efficient of $x^{r}$ in the expansion of $(1+x)^{20}$, then the value of $\underset{r=0}{\overset{20}{\sum\,}}r^{2}\left(Cr20\right)$ is equal to:

If the coefficient of $x^{10}$in the binomial expansion of $\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)^{60}$is $5^{k}l$, where $l,k\in\,N$and $l$is coprime to $5$, then $k$is equal to…