Sorry for the late posting (understatement).
Anyways, you can imagine this question as a group of 5 people and 2 other people - that group of 5 people will always sit next to each other, such that Pierre/Rosa/Thomas will be in the first, third, and fifth position of this group of 5 people. Now, you have 2 positions left, and since there are 4 people left, you can calculate this as \({4 \choose 2}=6\). Also, their are \(3!=6\) ways to have the 3 people moved. Multiplying writing these numbers, you get \(36\). Therefor, you have 36 ways to arrange the group of 5 (whom you can now treat as a single person!). On the other side of the round table, you have 2 people that can be arranged in \(2!=2\) ways, so you have \(36*2=72\), and then multiplying by 3 gives us \(72*3!=72*6=432\). However, since this is a round table, you must divide by 3 since you can rotate the table as
A
B C
or
B
C A
or
C
A B,
Giving us a total of 3 ways you overcounted the original \(432\), so you must divide by 3 to get \(\frac{432}{3}=\boxed{144} \).
(However, you divide by 3 because this question only asks for rotations, but other questions ask for reflections/flips, in which case divide by an additional 2.)
