Binomial Theore...

The coefficient of $$x^5$$ in the expansion of $$\dfrac{1+x^2}{a+x},|x| < 1$$, is 

If the ratio of the seventh term from the beginning of the binomial expansion of $${ \left( { 2 }^{ 1/3 }+\cfrac { 1 }{ { 3 }^{ 1/3 } }  \right)  }^{ x }$$ to the seventh term from its end is $$\cfrac { 1 }{ 6 } $$, then the value $$x$$ is

If $${ C }_{ 0 },{ C }_{ 1 },{ C }_{ 2 },.....{ C }_{ r }$$ are binomial coefficients in the expansion of $${(1+x)}^{n}$$ then
$${ C }_{ 1 }-\cfrac { { C }_{ 2 } }{ 2 } +\cfrac { { C }_{ 3 } }{ 3 } -\cfrac { { C }_{ 4 } }{ 4 } +....{ \left( -1 \right)  }^{ n-1 }\cfrac { { C }_{ n } }{ n } $$ equals

The coefficient of x in the expansion of $$(1-ax)^{-1}(1-bx)^{-1}(1-cx)^{-1}$$ is?

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

If $${C}_{r}$$ denotes the binomial coefficient $${ _{  }^{ n }{ C } }_{ r }$$ then $$\left( -1 \right) { C }_{ 0 }^{ 2 }+2{ C }_{ 1 }^{ 2 }+5{ C }_{ 2 }^{ 2 }+......(3n-1){ C }_{ n }^{ 2 }=\quad $$

The value of $$\cfrac { { C }_{ 0 } }{ 1.3 } -\cfrac { { C }_{ 1 } }{ 2.3 } +\cfrac { { C }_{ 2 } }{ 3.3 } -\cfrac { { C }_{ 3 } }{ 4.3 } +.....{ \left( -1 \right)  }^{ n }\cfrac { { C }_{ n } }{ (n+1).3 } $$ is

Find the term of the expansion of $$\displaystyle\, \left ( \sqrt[3]{x^{-2}} + x \right )^7$$ containing $$x$$ in the second power.

The numerically greatest terms in the expansion of $${ \left( 2x+5y \right)  }^{ 34 }$$ when $$x=3$$ & $$y=2$$ is

If $$T_0, T_1, T_2, .....T_n$$ represent the term in the expansion of $$(x+a)^n$$, then the value of $$(T_0- T_2+ T_4- T_6+ .....)^2+(T_1-   T_3+ T_5+.....)^2$$ is