Binomial Theore...

The number of terms in the expansion of $$\left [ (a+4b)^{3}(a-4b)^{3} \right ]^{2}$$ are

The number of terms in the expansion of 

$$(a_{1}+b_{1})(a_{2}+b_{2}).....(a_{n}+b_{n})$$

(i) The no.of distinct terms in the expansion of $$({x}_{1}+{x}_{2}+\ldots.+{x}_{{n}})^{3}$$ is $$n+2{c}_{3}$$

(ii) The no. of irrational terms in the expansion $$(2^{1/5}+3^{1/10})^{55}$$ is $$55$$

The sum of the coefficients of the middle terms of $$(1+{x})^{2{n}-1}$$ is

If $$x + y = 1$$, then $$\displaystyle \sum_{r=0}^{n}r^{n}C_{r}x^{r}.y^{n-r}=$$

In the expansion of $$\left ( 2.2^{\dfrac{2x}{3}}+2^{\dfrac{-x}{3}} \right )^{n}$$ the sum of the last four binomial coefficients exceeds the sum of the first three binomial coefficients by 20 and if the middle term is-the numerically largest term, then x belongs to

If $$x + y = 1$$ then $$\displaystyle \sum_{r=0}^{n}r^{2}$$  $$^{n}C_{r}x^{r}y^{n-r}$$

In the binomial expansion of $$(a - b)^{n}, n>5,$$ the sum of 5$$^{th}$$ and 6$$^{th}$$ terms is zero, then $$\displaystyle\frac{a}{b}$$ equals-

If the fourth term in the expansion of 

$$\left(\sqrt{x^\dfrac{1}{\log x+1}}+x^\dfrac{1}{12}\right)^{6}$$ is equal to $$200$$ and $$x>1$$, then $$x$$ is

If in the expansion of $$(\displaystyle \frac{1}{x}+x\tan x)^{5}$$, the ratio of $$4^{th}$$ term to the $$2^{nd}$$ term is $$\displaystyle \frac{2}{27}\pi^{4}$$, then the value of $${x}$$ can be