Coefficient of $x^{n-6}$ in the expansion $n\left[ {x - \left( {\frac{{^n{C_0}{ + ^n}{C_1}}}{{^n{C_0}}}} \right)} \right]\left[ {\frac{x}{2} - \left( {\frac{{^n{C_1}{ + ^n}{C_2}}}{{^n{C_1}}}} \right)} \right]\left[ {\frac{x}{3} - \left( {\frac{{^n{C_2}{ + ^n}{C_3}}}{{^n{C_2}}}} \right)} \right].....$ $ \left[ {\frac{x}{n} - \left( {\frac{{^n{C_{n - 1}}{ + ^n}{C_n}}}{{^n{C_{n - 1}}}}} \right)} \right]$ is equal to (where $n = n . (n -1) . (n -2).... 3.2.1$ )

The value of $\left( \begin{array}{l}30\\0\end{array} \right)\,\left( \begin{array}{l}30\\10\end{array} \right) - \left( \begin{array}{l}30\\1\end{array} \right)\,\left( \begin{array}{l}30\\11\end{array} \right)$ + $\left( \begin{array}{l}30\\2\end{array} \right)\,\left( \begin{array}{l}30\\12\end{array} \right) + ....... + \left( \begin{array}{l}30\\20\end{array} \right)\,\left( \begin{array}{l}30\\30\end{array} \right)$

Suppose $\sum \limits_{ r =0}^{2023} r ^{20023} C _{ r }=2023 \times \alpha \times 2^{2022}$. Then the value of $\alpha$ is $............$

If $\mathrm{b}$ is very small as compared to the value of $\mathrm{a}$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b}+\frac{1}{a-2 b}+\frac{1}{a-3 b} \ldots .+\frac{1}{a-n b}=\alpha n+\beta n^{2}+\gamma n^{3}$, then the value of $\gamma$ is:

Let $X =\left({ }^{10} C _1\right)^2+2\left({ }^{10} C _2\right)^2+3\left({ }^{10} C _3\right)^2+\ldots \ldots . .+10\left({ }^{10} C _{10}\right)^2$ where ${ }^{10} C _{ r }, r \in\{1,2, \ldots ., 10\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430} X$ is. . . . . . .

The value of $-{ }^{15} C _{1}+2 .{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots \ldots$ $-15 .{ }^{15} C _{15}+{ }^{14} C _{1}+{ }^{14} C _{3}+{ }^{14} C _{5}+\ldots .+{ }^{14} C _{11}$ is