In how many ways can we seat 6 people around a round table if Fred and Gwen and Scott insist on sitting next to each other? (Two seatings are considered equivalent if one is a rotation of the other.)

tie Fred Gwen and scott together in a line. there are 3! arrangements for this

Now put them in the circle, they determine the 'start' of the pattern

There are 3 more people which can be arranged in 3! ways

So the answer is 3!*3!