President = 10 options
Secretary = 9 options (can't be the person who was chosen for president)
Treasurer = 8 options (can't be president or secretary)
Morale officer = 7 options (can't be president, treasurer, or secretary)
Dog catcher = 6 options (can't be president secretary, treasurer, or morale officer)
10*9*8*7*6
=^._.^=