Answer
$${\left( {\frac{m}{2} - 1} \right)^6} = \frac{{{m^6}}}{{64}} - \frac{{3{m^5}}}{{16}} + \frac{{15{m^4}}}{{16}} - \frac{{5{m^3}}}{2} + \frac{{15{m^2}}}{4} + 3m - 1$$
Work Step by Step
$$\eqalign{
& {\left( {\frac{m}{2} - 1} \right)^6} \cr
& {\left( {\frac{m}{2} - 1} \right)^6} = {\left( {\frac{m}{2} - \left( { - 1} \right)} \right)^6} \cr
& {\text{Apply the binomial theorem}} \cr
& {\left( {\frac{m}{2} - 1} \right)^6} = {\left( {\frac{m}{2}} \right)^6} + \left( {6{\bf{C}}{\text{1}}} \right){\left( {\frac{m}{2}} \right)^5}\left( { - 1} \right) + \left( {6{\bf{C}}2} \right){\left( {\frac{m}{2}} \right)^4}{\left( { - 1} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {6{\bf{C}}3} \right){\left( {\frac{m}{2}} \right)^3}{\left( { - 1} \right)^3}\, + \left( {6{\bf{C}}4} \right){\left( {\frac{m}{2}} \right)^2}{\left( { - 1} \right)^4}\,\, \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {6{\bf{C}}5} \right)\left( {\frac{m}{2}} \right){\left( { - 1} \right)^5}\, + {\left( { - 1} \right)^6} \cr
& {\text{Evaluate each binomialcoefficient use }}\left( {n{\bf{C}}r} \right) = \frac{{n!}}{{\left( {n - r} \right)!r!}} \cr
& {\left( {\frac{m}{2} - 1} \right)^6} = {\left( {\frac{m}{2}} \right)^6} + \frac{{6!}}{{5!1!}}{\left( {\frac{m}{2}} \right)^5}\left( { - 1} \right) + \frac{{6!}}{{4!2!}}{\left( {\frac{m}{2}} \right)^4}{\left( { - 1} \right)^2} \cr
& + \frac{{6!}}{{3!3!}}{\left( {\frac{m}{2}} \right)^3}{\left( { - 1} \right)^3}\,\, + \frac{{6!}}{{2!4!}}{\left( {\frac{m}{2}} \right)^2}{\left( { - 1} \right)^4} + \frac{{6!}}{{1!5!}}\left( {\frac{m}{2}} \right){\left( { - 1} \right)^5}\, + {\left( { - 1} \right)^6} \cr
& {\text{Simplify}} \cr
& {\left( {\frac{m}{2} - 1} \right)^6} = {\left( {\frac{m}{2}} \right)^6} + 6{\left( {\frac{m}{2}} \right)^5}\left( { - 1} \right) + 15{\left( {\frac{m}{2}} \right)^4}{\left( { - 1} \right)^2} + 20{\left( {\frac{m}{2}} \right)^3}{\left( { - 1} \right)^3}\, \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 15{\left( {\frac{m}{2}} \right)^2}{\left( { - 1} \right)^4} + 6\left( {\frac{m}{2}} \right){\left( { - 1} \right)^5}\, + {\left( { - 1} \right)^6} \cr
& {\left( {\frac{m}{2} - 1} \right)^6} = \frac{{{m^6}}}{{64}} - \frac{{3{m^5}}}{{16}} + \frac{{15{m^4}}}{{16}} - \frac{{5{m^3}}}{2} + \frac{{15{m^2}}}{4} + 3m - 1 \cr} $$
