Binomial Theore...

$$\text{5}^{th}$$ term from the end in the expansion of $${\left( {\frac{{{x^3}}}{2} - \frac{2}{{{x^2}}}} \right)^{12}}$$ is :

If the $$rth$$ term in the expansion of $$\left(\dfrac{x}{3}-\dfrac{2}{x^{2}}\right)^{10}$$ contains $$x^{4}$$ then $$r$$ is equal to

The sum of the binomial coefficients in the expansion of $${ \left( { x }^{ -3/4 }+a{ x }^{ 5/4 } \right)  }^{ n }$$ lies between $$200$$ and $$400$$ and the term independent of $$x$$ equals $$448$$. The value of $$a$$ is

The sum of the series
$$\dfrac {2\left(\dfrac {n}{2}\right)!\left(\dfrac {n}{2}\right)!}{(n!)}[C^{2}_{0}-2C^{2}_{1}+3C^{2}_{2}...+(-1)^{n}(n+1)C^{2}_{n}]$$
where $$n$$ is an even positive integer, is equal to

Sum of coefficients of $${ x }^{ 2r }$$, $$r= 1,2,3,$$....... in $$(1+x{ ) }^{ n }$$ is

Coefficient of $$x^{n}$$ in expansion of $$\dfrac{(1+2x)^{2}}{(1-x)^{3}}$$ is

$$\begin{array} { l } { \text { Assertion } ( \mathrm { A } ) : \text { The expansion of } ( 1 + x ) ^ { n } = } \\ { C _ { 0 } + C _ { 1 } x + C _ { 2 } x ^ { 2 } + \ldots + C _ { n } x ^ { n } } \\ { \text { Reason (R): If } x = - 1 , \text { then the above expansion is } } \\ { \text { zero } } \end{array}$$

The coefficient of $$x^{99}$$ in $$(x+1)(x+3)(x+5).....(x+199)$$ is

The middle term in the expansion of $${\left(3x-\dfrac{{x}^{3}}{6}\right)}^{9}$$ is 

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