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

Question

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.

• The last question, Can someone tell me which type of problems I need to study to get a good knowledge on such problems?, is not so clear... – Τίμων Mar 12 '16 at 20:24
• thanks, @Timon, I did not understand it from the explanation which i quoted there they just identified 5 and multiplied by 7 due to rotation. Then I thought it is some new concept. Your answer clarified my doubts. thanks. – Kiran Mar 12 '16 at 20:29
• In the first paragraph of your question, it should read "... if and only if $7$ divides $k$." – N. F. Taussig Mar 12 '16 at 20:30
• The answer at that site is valid if the chairs are labeled. If they are not labeled, then you have to divide by $7$ to account for invariance under rotation. These problems are called circular permutations. – N. F. Taussig Mar 12 '16 at 20:36
• Reading the statement again gives me the impression that the author means that any seating rotation that does not place all the people in their original seats is considered a new seating arrangement. That suggests that either the seats are labeled or that they are viewed from a particular reference point, say the north end of the table. Usually in circular permutation problems, two seating arrangements are considered the same if people are seated in the same places relative to each other. However, here there is some external reference point, making the given answer valid. – N. F. Taussig Mar 12 '16 at 20:58

If I have not misunderstood your question, the number of ways that four of the seven places on the round table can be occupied is just the same as the number of ways you can choose $4$ objects (in this case, the occupied seats) among $7$ different objects (in this case, all the seats), which is $\displaystyle \binom{7}{4}=35$.