Find the coefficient of $$x^{2}$$ in the equation of $$(1+2x)^{6}(1-x)^{-1}$$

Solution

$${\left( {1 + 2x} \right)^6}{\left( {1 - x} \right)^{ - 1}}$$
$$ = {\left( {1 - x} \right)^{ - 1}}{\left( {1 + 2x} \right)^6}$$
$$ = \left( {1 + x + {x^2} + {x^3} + .......} \right)\left( {^6{C_0} \times {{\left( {2x} \right)}^0}{ + ^6}{C_1}\left( {2x} \right){ + ^6}{C_2}{{\left( {2x} \right)}^2}{ + ^6}{C_3}{{\left( {2x} \right)}^3}{ + ^6}{C_4}{{\left( {2x} \right)}^4}{ + ^6}{C_5}{{\left( {2x} \right)}^5}{ + ^6}{C_6}{{\left( {2x} \right)}^6}} \right)$$
$$ = \left( {1 + x + {x^2} + {x^3} + .......} \right)\left( {1 + 12x + 60{x^2} + 160{x^3} + 240{x^4} + 192{x^4} + 64{x^6}} \right)$$
So,
    Coefficient of $${x^2}=60+12+1$$
                                 $$=73$$