The coefficent of $x^7$ in the expansion of ${\left( {1 – x – {x^2} + {x^3}} \right)^6}$ is

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 :

If ${(1 + x – 2{x^2})^6} = 1 + {a_1}x + {a_2}{x^2} + …. + {a_{12}}{x^{12}}$, then the expression ${a_2} + {a_4} + {a_6} + …. + {a_{12}}$ has the value

$\sum\limits_{n = 0}^4 {{{\left( {1009 – 2n} \right)}^4}\left( \begin{gathered}
  4 \hfill \\
  n \hfill \\ 
\end{gathered}  \right)} {\left( { – 1} \right)^n}$   is

If the expansion in powers of $x$ of the function  $\frac{1}{{\left( {1 – ax} \right)\left( {1 – bx} \right)}}$ is ${a_0} + {a_1}x + {a_2}{x^2} + \;{a_3}{x^3} + \; \ldots……$ then  ${a_n}$ is