In how many ways can 8 people sit around a round table if 3 of the people -- Pierre, Rosa, and Thomas -- all want to sit together?  (Two seatings are considered the same if one is a rotation of the other.)
The problem asks about the number of ways to arrange 8 people around a round table, but there is a constraint that Pierre, Rosa, and Thomas must sit together. One way to approach this problem is to think of Pierre, Rosa, and Thomas as a single unit, and then arrange the other 5 people and the unit around the table. How many ways can we arrange 6 objects around a round table? Well, if we fix one object as the reference point, then the other 5 objects can be arranged in 5! ways. So the answer is 5!. However, we are not done yet, because within the unit of Pierre, Rosa, and Thomas, they can also switch places. How many ways can they do that? They can do that in 3! ways, since there are 3 people in the unit. So, for each arrangement of the 6 objects, there are 3! ways to arrange the 3 people in the unit. Therefore, the total number of ways to arrange 8 people around the round table with the constraint is 5! * 3!. Using a calculator or mental arithmetic, we can find that 5! * 3! = 120 * 6 = 720. 