Two men, Adam and Charles, and two women, Beth and Diana, sit at a table where there are seven places for them to sit down. Two people are sitting next to each other if they occupy consecutive chairs. A non-trivial rotation defines a different seating arrangement, meaning that if all four people rotate their positions by moving k chairs to the right, it is the same way for them to be seated if and only if k divides 7.
Determine the number of ways that four of the seven places on the round table can be occupied by people, (ignoring the names of the people seated on them).
I got the following explanation (here) which I am quoting below
PPPPxxx PPPxPxx PPPxxPx PPxPPxx PPxPxPx total : 5*7=35
Do we have any standard formula to approach this problem? How we get 35 as answer by easily generating the pattern as given above?
Can someone tell me which type of problems I need to study to get a good knowledge on such problems.