The co-efficient of $$x^6$$ in $$\left\{ {{{\left( {1 + x} \right)}^6} + {{\left( {1 + x} \right)}^7} + ..... + {{\left( {1 + x} \right)}^{15}}} \right\}$$        is

Solution :
Expression $$ = \frac{{{{\left( {1 + x} \right)}^6}\left\{ {1 - {{\left( {1 + x} \right)}^{10}}} \right\}}}{{1 - \left( {1 + x} \right)}} = \frac{{{{\left( {1 + x} \right)}^{16}} - {{\left( {1 + x} \right)}^6}}}{x}$$
∴ the required co-efficient
= the co-efficient of $$x^7$$ in $$\left\{ {{{\left( {1 + x} \right)}^{16}} - {{\left( {1 + x} \right)}^6}} \right\}$$
$$ = {\,^{16}}{C_7} = {\,^{16}}{C_{16 - 7}}.$$