Answer
$32,760$ different ways
Work Step by Step
First we ask, does order of selection matter?
It does, because being selected as President is different to being selected as Secretary.
Order matters, we use permutations.
Permutations Formula:
The number of permutations possible if $r$ items are taken from $n$ items is ${}_{n}P_{r}=\displaystyle \frac{n!}{(n-r)!}$
The number of officers chosen are r=4 from a total of n=15
${}_{15}P_{4}=\displaystyle \frac{15!}{(15-4)!}=\frac{15!}{11!}\qquad $... apply $n!=n(n-1)!=n(n-1)(n-2)!=...$
$=\displaystyle \frac{15\times 14\times 13\times 12\times 11!}{11!}$
$=15\times 14\times 13\times 12$
$=32,760$