In how many ways can we seat 6 people around a table if Al and Joe won't sit next to each other? (Two seatings are the same if one is a rotation of the other.)

Place Joe in any chair

Then Al can occupy any of 3 chairs (except the two next to Joe) = 3 ways

And the other people can sit in 4! = 24 ways for each of these arrangements

So 3 * 24 = 72 ways