# Sitting n families around a circular table with a condition

How many ways are there for sitting n families around a circular table. Each family is a mother a father and a child.

Condition: The mother and father of each family should be sitting next to each other while children are not allowed to sit next to each other.

Thanks

• What do you regard as distinct ways? Would you say the answer was 1 for $n=1$? – almagest Apr 5 '16 at 13:37
• For n=1 the answer would be 2 as the conditions do not apply here and the number of arrangement of 3 people around a circular table is (3-1)!=2 – Zoltán Apr 5 '16 at 13:55

I assume you regard arrangements obtained by rotations and reflections as equivalent.

Start with the location of $M_1$ and choose the direction by taking the next person as $F_1$. Then there are $(n-1)!2^{n-1}$ ways of arranging the other parents and $n!$ ways of fitting the children in the $n$ gaps between them. So total $n!(n-1)!2^{n-1}$.

• Arranging n couples of F's and M's should be (n-1)! multiplied by 2^n as each couple could be sitted as MF or FM. Now we need to arrange the n children and that gives us n! options. To conclude, (n-1)!* 2^n * n!. Could you spot the mistake I am making? – Zoltán Apr 5 '16 at 13:51
• need divide for circle rotations – miniparser Apr 5 '16 at 13:58
• @miniparser I see. Could you elaborate more please? – Zoltán Apr 5 '16 at 14:00
• @Tamir The two questions of definition are: (1) do you regard 1,2,3,4 as the same arrangement as 2,3,4,1 (ie differ only in rotation); (2) do you regard 1, 2, 3, 4 as the same arrangement as 1,4,3,2 (ie going the opposite way round the circle). I assumed the answer was yes in both cases. If your answer to (2) is No, then you need to double the number. – almagest Apr 5 '16 at 14:05
• I think divide by 3n – miniparser Apr 5 '16 at 14:10

Hint:-

Let us name the mother=$M$,father=$F$ and child=$C$.

As the mother and father sits together,group one family into two parts-$(M,F);(C)$.(How many groups can be formed out of n families if one family breaks up into $2$ groups?).

Now,arrange n number of $(M,F)$ and $(C)$'s around the table by using circular permutation formula that x number of objects can be arranged into $(x-1)!$ ways.