jmc

The best way to approach this problem is to use complementary counting. We already know that there are $20\times 19\times 18$ ways to choose the 3 officers if we ignore the restriction about Alex and Bob. So now we want to count the number of ways that both Alex and Bob serve as officers.

For this, we will use constructive counting. We need to pick an office for Alex, then pick an office for Bob, then put someone in the last office. We have 3 choices for an office for Alex, either President, VP, or Treasurer. Once we pick an office for Alex, we have 2 offices left from which to choose an office for Bob.

Once we have both Alex and Bob installed in offices, we have 18 members left in the club to pick from for the remaining vacant office. So there are $3\times 2\times 18$ ways to pick officers such that Alex and Bob are both in an office. Remember that these are the cases that we want to exclude, so to finish the problem we subtract these cases from the total number of cases. Hence the answer is: $$(20\times 19\times 18)-(3\times 2\times 18)=((20\times 19)-6)\times 18=374\times 18=\boxed{6732}.$$