Answer
$$729{x^6} - 2916{x^5}y + 4860{x^4}{y^2} - 4320{x^3}{y^3} + 2160{x^2}{y^4} - 576x{y^5}\, + 64{y^6}$$
Work Step by Step
$$\eqalign{
& {\left( {3x - 2y} \right)^6} \cr
& {\left( {3x - 2y} \right)^6} = {\left( {3x + \left( { - 2y} \right)} \right)^6} \cr
& {\text{Apply the binomial theorem}} \cr
& {\left( {3x - 2y} \right)^6} = {\left( {3x} \right)^6} + \left( {6{\bf{C}}{\text{1}}} \right){\left( {3x} \right)^5}\left( { - 2y} \right) + \left( {6{\bf{C}}2} \right){\left( {3x} \right)^4}{\left( { - 2y} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {6{\bf{C}}3} \right){\left( {3x} \right)^3}{\left( { - 2y} \right)^3}\, + \left( {6{\bf{C}}4} \right){\left( {3x} \right)^2}{\left( { - 2y} \right)^4} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {6{\bf{C}}5} \right)\left( {3x} \right){\left( { - 2y} \right)^5}\, + {\left( { - 2y} \right)^6} \cr
& {\text{Evaluate each binomialcoefficient use }}\left( {c{\bf{C}}r} \right) = \frac{{n!}}{{\left( {n - r} \right)!r!}} \cr
& {\left( {3x - 2y} \right)^6} = {\left( {3x} \right)^6} + \frac{{6!}}{{5!1!}}{\left( {3x} \right)^5}\left( { - 2y} \right) + \frac{{6!}}{{4!2!}}{\left( {3x} \right)^4}{\left( { - 2y} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{6!}}{{3!3!}}{\left( {3x} \right)^3}{\left( { - 2y} \right)^3}\, + \frac{{6!}}{{2!4!}}{\left( {3x} \right)^2}{\left( { - 2y} \right)^4} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{6!}}{{1!5!}}\left( {3x} \right){\left( { - 2y} \right)^5}\, + {\left( { - 2y} \right)^6} \cr
& {\text{Simplify}} \cr
& {\left( {3x - 2y} \right)^6} = {\left( {3x} \right)^6} + 6{\left( {3x} \right)^5}\left( { - 2y} \right) + 15{\left( {3x} \right)^4}{\left( { - 2y} \right)^2} + 20{\left( {3x} \right)^3}{\left( { - 2y} \right)^3}\, \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 15{\left( {3x} \right)^2}{\left( { - 2y} \right)^4} + 6\left( {3x} \right){\left( { - 2y} \right)^5}\, + {\left( { - 2y} \right)^6} \cr
& {\left( {3x - 2y} \right)^6} = 729{x^6} + 6\left( {243{x^5}} \right)\left( { - 2y} \right) + 15\left( {81{x^4}} \right)\left( {4{y^2}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 20\left( {27{x^3}} \right)\left( { - 8{y^3}} \right)\,\, + 15\left( {9{x^2}} \right)\left( {4{y^4}} \right) + \left( {18x} \right)\left( { - 32{y^5}} \right)\, + 64{y^6} \cr
& {\left( {3x - 2y} \right)^6} \cr
& = 729{x^6} - 2916{x^5}y + 4860{x^4}{y^2} - 4320{x^3}{y^3} + 2160{x^2}{y^4} - 576x{y^5}\, + 64{y^6} \cr} $$