We have 14 members and we need to arrange them in three positions, then we need to use a permutation.
A permutation is given as:
[tex]_nP_r=\frac{n!}{(n-r)!}[/tex]In this case we have:
[tex]\begin{gathered} _{14}P_3=\frac{14!}{(14-3)!} \\ =\frac{14!}{11!} \\ =\frac{14\cdot13\cdot12\cdot11!}{11!} \\ =14\cdot13\cdot12 \\ =2184 \end{gathered}[/tex]Therefore there are 2184 ways to choose the officers.
