# Seating $3$ couples around a circular table with husband and wife opposite

Find the number of ways in which three couples can be seated around a circular table such that husband and wife are always diametrically opposite to each other? How do I approach this problem? I think the answer should be $$2!*2!*2!=8$$ but my friend told me that the answer is $$3*2!*2!*2!=24$$. I think it should be $$8$$ only. Can anyone help? Pleaseđź™Źđź™Ź. Thank you

Label the people as $$M_1, M_2, M_3, W_1, W_2, W_3$$, where $$M_k$$ and $$W_k$$ must be seated across from each other, where $$k \in \{1,2,3\}.$$

Then, the problem reduces to determining how many ways there are of seating $$M_1, M_2, M_3$$, because once they are seated, the seating of $$W_1, W_2, W_3$$ is then determined.

Since the table is round, without loss of generality, $$M_1$$ sits in seat-1, which implies that seat-4 must be reserved for $$W_1$$.

There are therefore $$4$$ choices for $$M_2$$. Once $$M_1$$ and $$M_2$$ are seated, you now have $$2$$ seats that are reserved, for $$W_1$$ and $$W_2$$.

Therefore, once $$M_1$$ and $$M_2$$ are seated there are $$4$$ seats taken. Therefore, there are then $$2$$ choices remaining for the seating of $$M_3$$.

Therefore, the final computation is

$$4 \times 2 = 8 ~~\text{ways}.$$

• But $M_k$ and $W_k$ can switch their seats so wouldn't the answer be $8\times2=16$ ? Commented Oct 29, 2021 at 20:38
• @Soheil No. Under the assumption that the table is round, seat-1 is reserved for $M_1$. Then, the problem reduces to seating $M_2$ and $M_3$. Any variation in the seating that you might construct by having a married couple switch places has already been counted, as long as you go to the seat where $M_1$ is sitting, label that as seat-1, and then (for example), label the remaining seats in clockwise order (when the table is viewed from above). Commented Oct 29, 2021 at 20:41
• @Soheil Proof is in the pudding. Try to construct a seating arrangement that you think that I have overlooked, and then see whether that arrangement is actually part of the $8$ arrangements that I have counted. Commented Oct 29, 2021 at 20:43
• If for some reason you insist that rotations are distinct from each other, then you have to multiply by 6 to account for them. I believe your counting scheme already accounts for reflections, so we need not adjust for them. Commented Oct 30, 2021 at 4:33
• @Kevin Agreed, but in my experience with such math problems, typically, the whole point of specifying circular-table-seating is the unspoken premise that rotations are not distinct from each other. Otherwise, such a problem is generally presented as the people being seated in a row, where each husband-wife pairing must have exactly $2$ other seats between them. Commented Oct 30, 2021 at 5:19