Co-efficient of $\alpha ^t$ in the expansion of,

$(\alpha + p)^{m – 1} + (\alpha + p)^{m – 2} (\alpha + q) + (\alpha + p)^{m – 3} (\alpha + q)^2 + …… (\alpha + q)^{m – 1}$

where $\alpha \ne – q$ and $p \ne q$ is :

Let $m, n \in N$ and $\operatorname{gcd}(2, n)=1$. If $30\left(\begin{array}{l}30 \\ 0\end{array}\right)+29\left(\begin{array}{l}30 \\ 1\end{array}\right)+\ldots+2\left(\begin{array}{l}30 \\ 28\end{array}\right)+1\left(\begin{array}{l}30 \\ 29\end{array}\right)= n .2^{ m }$ then $n + m$ is equal to (Here $\left.\left(\begin{array}{l} n \\ k \end{array}\right)={ }^{ n } C _{ k }\right)$

In the polynomial $(x – 1)(x – 2)(x – 3)………….(x – 100),$ the coefficient of ${x^{99}}$ is

$\left( {\left( {\begin{array}{*{20}{c}}
{21}\\
1
\end{array}} \right) – \left( {\begin{array}{*{20}{c}}
{10}\\
1
\end{array}} \right)} \right) + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
2
\end{array}} \right) – \left( {\begin{array}{*{20}{c}}
{10}\\
2
\end{array}} \right)} \right)$$ + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
3
\end{array}} \right) – \left( {\begin{array}{*{20}{c}}
{10}\\
3
\end{array}} \right)} \right) + \;.\;.\;.$$ + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
{10}
\end{array}} \right) – \left( {\begin{array}{*{20}{c}}
{10}\\
{10}
\end{array}} \right)} \right) = $

${n^n}{\left( {\frac{{n + 1}}{2}} \right)^{2n}}$ is