Answer
$32760$ possibilities.
Work Step by Step
If we want to choose $k$ elements out of $n$ regarding the order, not allowing repetition, we can do this in $_{n}P_k=\frac{n!}{(n-k)!}$ ways.
The order matters here when choosing the four officers, thus we have to use permutations. We have $15$ people for $4$ spots, thus $_{15}P_{4}=\frac{15!}{(15-4)!}=15\cdot14\cdot13\cdot12=32760$ possibilities.
