Binomial Theore...

The coefficient of $$x^{m}$$ in $$(1 + x)^{p} + (1 + x)^{p + 1} + .... + (1 + x)^{n}, p\leq m\leq n$$ is

The sum of the binomial coefficients of $$\left[2x+\cfrac{1}{x}\right]^n$$ is equal to $$256$$. The constant term in the expansion is 

The coefficient of $$x^{9}$$ in the expansion of (1+x)$$(1+x^{2})(1+x^{3})....(1+x^{100})$$ is____.

The first integral term in the expansion of $$(\sqrt{3}+\sqrt[3]{2})^{9}$$, is its

The sum of the coefficients in the expansion of $${ (1+x-{ 3x }^{ 2 }) }^{ 171 }$$ is 

Co-efficient of $$x^{n-1}$$ in the expansion of, $$(x+3)^{n}+(x+3)^{n-1}(x+2)+(x+3)^{n-2}(x+2)^{2}+...+(x+2)^{n}$$ is:

In the expansion of $$(x+y+z)^{10}$$

If in the expansion of  $$\left( 2 ^ { x } + \dfrac { 1 } { 4 ^ { x } } \right) ^ { n } , T _ { 3 } = 7 T _ { 2 }$$  and sum of the binomial coefficients of second and third terms is  $$36 ,$$  then the value of  $$x$$  is -

The first three terms in the expansion of $$(1 - ax)^{n}$$, where $$n$$ is positive integer are, $$1 - 4x + 7x^{2}$$, then the value of $$a$$

The value of x for  which the sixth term in the expansion of 
$$\left [ 2^{\log_{2}{\sqrt{9^{x-1}+7}}}+\dfrac{1}{2^{\frac{1}{5}\log_{2}{(3^{x-1}+1)}}} \right ]^{7}$$ is equal to 84 is