There is more than one way to think of this. I think that the best way to explain it is to begin with first principles. There are 4 distinct offices to be filled, and they are not interchangeable. Therefore order matters.
Consider beginning to select these officers one at a time, beginning with president. There are 25 people to choose from that could potentially be president. Once that person has been selected, he/she cannot be selected for another office. So when it comes to selecting the vice-president, any of the 24 remaining people could be selected for that office. When VP has been decided, then there are 23 people left, from which we may choose a secretary. Once that is done, then there are 22 left, from which to choose a vice secretary.
Each of these events are independent and happen together, so by the fundamental counting principle, we multiply each possible outcome 25 • 24 • 23 • 22 = 303,600.
