The co-efficient of $${x^{15}}$$ in the product $$\left( {1 - x} \right)\left( {1 - 2x} \right)\left( {1 - {2^2} \cdot x} \right)\left( {1 - {2^3} \cdot x} \right).....\left( {1 - {2^{15}} \cdot x} \right)$$           is equal to

Solution :
Product $$ = \left( { - 1} \right)\left( { - 2} \right)\left( { - {2^2}} \right).....\left( { - {2^{15}}} \right)\left( {x - 1} \right)\left( {x - \frac{1}{2}} \right)\left( {x - \frac{1}{{{2^2}}}} \right)\left( {x - \frac{1}{{{2^3}}}} \right).....\left( {x - \frac{1}{{{2^{15}}}}} \right)$$
$$ = {2^{1 + 2 + 3 + ..... + 15}}\left( {x - 1} \right)\left( {x - \frac{1}{2}} \right)\left( {x - \frac{1}{{{2^2}}}} \right).....\left( {x - \frac{1}{{{2^{15}}}}} \right)$$
∴ co-efficient of $${x^{15}} = {2^{1 + 2 + 3 + ..... + 15}}.\left( { - 1 - \frac{1}{2} - \frac{1}{{{2^2}}} - ..... - \frac{1}{{{2^{15}}}}} \right).$$