In how many ways can $7$ people sit around a round table if no two of the $3$ people Pierre, Rosa, and Thomas can sit next to each other? (Seating arrangements which are rotations of each other are treated as the same.)
Let's first seat Pierre, Rosa, and Thomas. Yes, and let's make sure that they are not sitting next to each other. That means that between any two of them, there must be at least one seat. First, let's take the case where Rosa is two seats from Pierre. That means that she is either one seat to the left or one seat to the right of Pierre. So for this case, there are two seats for Thomas to sit in. For case 1), there are 2 seats for Thomas and then 4! for the other people. For case 2), there are 1 seat for Thomas and then 4! for the other people. Wait, we're not done yet. We need to add the two cases together. Right. $96+48=144$. Ok, so there are $144$ ways to seat the 7 people.

