Binomial Theorem for Positive Integral Index

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{15}C_5$

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{20}C_7$

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{10}C_9$

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{8}C_6$

State whether the following binomial coefficient exist or not. If exists, find the value.

${}^{12}C_8$

If the digits at ten's and hundred's places in ${\left(11\right)}^{2016}$ are $x$ and $y$ respectively, then the ordered pair $\left(x,y\right)$ is equal to

If the coefficient of$7^{\mathrm{th}}$ and ${13}^{\mathrm{th}}$ terms in the expansion of ${\left(1+x\right)}^n$, $\left(n\in N\right)$ are equal, then $n$ is equal to

In the expansion of ${\left(1+x\right)}^{15}+\mathrm{ }{\left(1+x\right)}^{16}+\mathrm{ }{\left(1+x\right)}^{17}+...+\mathrm{ }{\left(1+x\right)}^{30}$, the coefficient of $x^{10}$ is

The coefficient of $x^4$ in the expansion of ${\left(\frac{x}{2}-\frac{3}{x^2}\right)}^{10}$ is

The coefficient of $x^{\mathit{n}}$ in the expansion of ${\left(1+x\right)}^{2n}$ and ${\left(1+x\right)}^{2n-1}$ are in the ratio

Given positive integers $r>1, n>2$ and the coefficient of $3r^{th} \& {\left(r+2\right)}^{th}$ terms in the binomial expansion of ${\left(1+x\right)}^{2n}$ are equal. Then,

In the expansion of ${\left(x-\frac{1}{x}\right)}^6$, the coefficient of $x^0$ is

If ${}^{n-1}{C}_r=\left(k^2-3\right){·}^n{C}_{r+1},$ then the value of $k$ lies in

If $A$ and $B$ are coefficient of $x^{\mathit{n}}$ in the expansions of ${\left(1+x\right)}^{2n}$ and ${\left(1+x\right)}^{2n-1}$ respectively, then $\frac{A}{B}$ equals

The value of $\left({}^{21}{C}_1{-}^{10}{C}_1\right)+\left({}^{21}{C}_2{-}^{10}{C}_2\right)+\left({}^{21}{C}_3{-}^{10}{C}_3\right)$$+\left({}^{21}{C}_4{-}^{10}{C}_4\right)+\dots +\left({}^{21}{C}_{10}{-}^{10}{C}_{10}\right)$is

The number of terms in ${\left({\text{x}}^3+1+\frac{1}{{\text{x}}^3}\right)}^{100}$ is -

The total number of dissimilar  terms in the expansion of ${\left(x+a\right)}^{100}+{\left(x-a\right)}^{100}$ after simplification is

$x^5+10x^4 a+40x^3 a^2+80x^2 a^3+80 x{a}^4+32 a^5$ is equal to

$6^{th}$ term in expansion of ${\left(2x^2-\frac{1}{3x^2}\right)}^{10}$ is

If ${\left(27\right)}^{999}$ is divided by $7$, then the remainder is