# permutations around a round table with labelled seats

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 these four people can be seated so that every man is next to a woman and every woman is next to a man.

Answer is 224 . Here is the link with (source with explanation).

But my answer is 252. My calculations are given below.

place adam in any seat(7)
select any woman and place any of the adjacent seat (2*2)
place a man any of the 3 far away seats(3)
seat remaining woman in adjacent seat(2)
so 7*2*2*3*2= 168

select any woman and place any of the adjacent seat (2*2)
place a man in adjacent seat to the woman(1)
seat remaining woman in adjacent seat(2)
so 7*2*2*1*2= 56

select any woman and place any of the adjacent seat (2*2)
place a man in adjacent seat to the man(1)
seat remaining woman in adjacent seat(1)
so 7*2*2*1*1= 28

So total = 168+ 56+28 = 252


Can someone help me to figure out whether any of this answer is right? If my answer is wrong, please help to understand why it has gone wrong and how correct answer can be reached.

The explanation given in the site derives the answer as 224 but it is in a different approach than mine. Is that correct?

The correct answer is $224$. Your calculations are almost fine

Place Adam in any seat ($7$ ways)
Select any woman and place her in any of the adjacent seat ($2\cdot 2$ ways)
Place the other man in any of the $3$ non-adjacent seats ($3$ ways)
Place the remaining woman in any adjacent seat ($2$ ways)
Total: $7\cdot 2 \cdot 2\cdot 3\cdot 2= 168$ ways.

Place Adam in any seat ($7$ ways)
Select any woman and place her in any of the adjacent seat ($2\cdot 2$ ways)
Place the other man in the adjacent seat to the woman ($1$ way)
Place the remaining woman in the adjacent seat ($1$ way)
Total: $7\cdot 2\cdot 2\cdot 1\cdot 1= 28$ ways.

Place Adam in any seat ($7$ ways)
Select any woman and place her in any of the adjacent seat ($2\cdot 2$ ways)
Place the other man in the adjacent seat to the man ($1$ way)
Place the remaining woman in the adjacent seat ($1$ way)
Total: $7\cdot 2\cdot2\cdot1\cdot1= 28$ ways.

Total: $224$

As the last two blocks are just the same, you can improve your explanation by merging them.

• for the second block, adam, woman,man and then second woman can seat near to adam or the other man. so 2 ways, right? – Kiran Mar 13 '16 at 21:37
• Nope. The latter case is when the second woman is placed near Adam, and you have already counted that situation. – Τίμων Mar 13 '16 at 21:44