to each position? no person can hold more than one position.

This is a permutation problem. There are 15 different possibilities for the president. Once the president is selected, there are now only 14 different possibilities for vice-president. And once the vice-president is selected, there are only 13 different possibilities for the secretary-treasurer. So it is:

15 * 14 * 13 = 2730

15 * 14 * 13 = 2730