Binomial Theore...

If $$C_0, C_1, C_2, C_3, .... $$ are binomial coefficients in the expansion of $$ ( 1+x)^n $$ , then $$ \dfrac {C_0}{3} - \dfrac {C_1}{4} + \dfrac{}{} - \dfrac{}{} + ... $$ is equal to :

The value of $$r$$ for which the coefficients of $$(r-5)$$th and $$(3r+1)$$th terms in the expansion of $${(1+x)}^{1/2}$$ are equal, is

If $$(1 + x + x^{2})^{n} = 1 + a_{1}x + a_{2}x^{2} + ... + a_{2n}x^{2n}$$, then $$2a_{1} - 3a_{2} + ... -(2n + 1)a_{2n}$$ is equal to

The middle term in the expansion of $$\left( \dfrac{10}{x} + \dfrac{x}{10} \right )^{10}$$ is

If the term free from $$x$$ in the expansion of $$\left (\sqrt {x} - \dfrac {k}{x^{2}}\right )^{10}$$ is $$405$$, then the value of $$k$$ is

If $${ C }_{ 0 },{ C }_{ 1 },{ C }_{ 2 },\dots ,{ C }_{ 15 }$$ are binomial coefficients in $${ \left( 1+x \right)  }^{ 15 }$$, then $$\dfrac { { C }_{ 1 } }{ { C }_{ 0 } } +2\dfrac { { C }_{ 2 } }{ { C }_{ 1 } } +3\dfrac { { C }_{ 3 } }{ { C }_{ 2 } } +\cdots +15\dfrac { { C }_{ 15 } }{ { C }_{ 14 } } $$ is equal to

Sum of coefficients of the last $$6$$ terms in the expansion of $${ \left( 1+x \right)  }^{ 11 }$$ when the expansion is in ascending powers of $$x$$, is

The middle term in the expansion of $${ \left( 1+x \right)  }^{ 2n }$$ is

The value of $$\begin{pmatrix} 50 \\ 0 \end{pmatrix}\, \begin{pmatrix} 50 \\ 1 \end{pmatrix} + \begin{pmatrix} 50 \\ 1 \end{pmatrix}\, \begin{pmatrix} 50 \\ 2 \end{pmatrix}+........+\begin{pmatrix} 50 \\ 49 \end{pmatrix}\, \begin{pmatrix} 50 \\ 50 \end{pmatrix}$$ is , where $$^nC_r=\begin{pmatrix} n \\ r \end{pmatrix}$$

If $$n\epsilon N$$ and $$(1 + 4x + 4x^{2})^{n} = \displaystyle \sum_{r = 0}^{r = 2n} a_{r}x^{r}$$ then value of $$2\displaystyle \sum_{r = 0}^{n}a_{2r}$$ equals