Answer
The total number of ways in which the offices can be filled is 32,760 ways.
Work Step by Step
We know that the order in which the four officers are selected makes a difference as the designation for all the four officers would be different.
The ordered arrangement in which the order of the arrangement makes a difference is solved using the concept of permutations.
Four officers are to be selected from a club of fifteen members. So, $ n=15,r=4$.
Thus,
$\begin{align}
& _{15}{{P}_{4}}=\frac{15!}{\left( 15-4 \right)!} \\
& =\frac{15!}{11!} \\
& =\frac{15\times 14\times 13\times 12\times 11!}{11!} \\
& =32,760
\end{align}$
