OK, Cal....you ask a good question....!!!!
Let's look at something more simple
Suppose we only had 4 members A, B C D
Let A/B A/C A/D B/C B/D C/D represent president / vice-president
Note that this is C(4,2) = 4!
_______ = 6
(4 - 2)! 2!
But....what we have as actually a permutation because A/B isn't the same as B /A
So we actually have P(4,2) = 12
So your answer is a combination C(20,2) = 190
But we should have a permutation = P(20, 2) = 380...... this is the same result as 20 * 19