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.)

There are 48 ways to do this.

We will have 7 identical chairs around the table.

We need to seat Pierre, Rosa and Thomas so that no two of them are together.  This basically means alternating them between the other people at the table. Let Pierre sit first. Each seat is identical so he sits in one way. Now each seat is distinct relative to Pierre. There are 4 seats identified for the other members of the group and 2 for the Rosa and Thomas. The 4 other members can occupy the 4 distinct seats in 4! ways and Rosa and Thomas can occupy the 2 distinct seats in 2! ways.  This gives us 4!*2!=4(3)(2)(1)(2)(1)=48.