Binomial Theore...

The ratio of  coefficient  of $$x^{3}andx^{4}$$ in expansion $$(1+x)^{12}$$is:

The first $$3$$ terms in the expansion of $$(1+ax)^{n}(n\neq 0)$$ are $$1, 6x$$ and $$16x^{2}$$. Then the value of $$a$$ and $$n$$ are respectively 

For $$r=0,1,2,,....10$$ let $${A}_{r},{B}_{r}$$ and $${C}_{r}$$ denote respectively the coefficient of $${x}^{r}$$ in the expansions of $${(1+x)}^{10},{(1+x)}^{20}$$ and $${(1+x)}^{30}$$. Then $$\sum _{ r=1 }^{ 10 }{ { A }_{ r }\left( { B }_{ 10 }{ B }_{ r }-{ C }_{ 10 }{ A }_{ r } \right)  } $$ is equal to

The numerical value of middle terms in $${ \left( 1-\cfrac { 1 }{ x }  \right)  }^{ n }{(1-x)}^{n}$$ is

The value of
$$\left( { _{  }^{ 7 }{ C } }_{ 0 }+{ _{  }^{ 7 }{ C } }_{ 1 } \right) +\left( { _{  }^{ 7 }{ C } }_{ 1 }+{ _{  }^{ 7 }{ C } }_{ 2 } \right) +.....\left( { _{  }^{ 7 }{ C } }_{ 6 }+{ _{  }^{ 7 }{ C } }_{ 7 } \right) $$ is

The $$r$$th term of series $$2\dfrac{1}{2} + 1\dfrac{7}{{13}} + 1\dfrac{1}{9} + \dfrac{{20}}{{23}} + .....$$ is

If $$r$$th term is middle term in $${ \left( { x }^{ 2 }-\cfrac { 1 }{ 2x }  \right)  }^{ 20 }$$ then $$(r+3)$$th term is:

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

If the constant term in the expansion of $$\left(x^{2}-\dfrac{1}{x}\right)^{n}$$ is $$15$$ then the value of $$n$$ is

If $$(1-x-x^2)^{20}$$ = $$\sum _{ r=0 }^{ 40 }{ a_4,x^x },$$ then 
$$a_1+3a_3+5a_5+........+39a_{39}=$$