Find the number of ways when all men are adjacent to their wives.

Question

One British and four French men and their wives are to be seated on a round table.If $m$ denotes the number of ways in which each French man is seated adjacent to his wife and $n$ denotes the number of ways when all men are adjacent to their wives.Find $m$ and $n$.

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While finding $m$,i fixed the position of British man,and French men can be seated in $4!$ ways.But i cannot judge how to sit their wives(whether on their left or their right or either side ) and how to count total sitting positions.Same difficulty in counting $n$.

Please help me.

Answer 1

We can fix the British man's position. To find $m$, we consider each French couple as a unit and the Brit's wife as a unit. Thus there are $5!$ ways to arrange the units. But there are $4$ units of $2$ people each and $1$ unit of only $1$ person. For each $2$-piece unit, there are $2!$ arrangements of the people within that unit. So the final answer is $m=5!\cdot(2!)^4$.

To find $n$, we first consider where the Brit's wife can be sitting. There are 2 places adjacent to him, so that's 2 spots for her to choose from. Now there are 4 units of 2 people each to be arranged after the Brit and his wife are seated. The final answer, then, is $n=2\cdot4!\cdot(2!)^4$.

Answer 2

After seating the British lady anywhere, seat the French couples together in $8$ chairs in $4!*2^4$ ways. (One such pattern given below, L = British lady).

L _ Ff _ fF _ fF _ Ff _ (L)

There are 5 ways topush in a chair to seat the British man, so $m = 5*(4!*2^4)$

If the British couple also need to be together, there are only 2 ways the British man can be seated, so $n = 2*(4!*2^4)$

Note: This takes seats as unnumbered. For numbered seats, multiply both figures by $10$