Question

In how many ways can four couples be seated around a circular table such that people of same gender do not sit in adjacent positions and exactly one of the four couples sit in adjacent positions?

• A. $48$
• B. $36$
• C. $96$
• D. $24$

Answer 1

Answer: (A)

$48$

Lets arrange women first and then the men.

Number of arranging $4$ women on a round table $=3!$

Now men have to be seated in between the women in the gaps.

Only one couple can sit together so no of ways of selecting this couple $={}^4{C}_1=4$

Then the selected man has two places to sit besides his wife that is on the two sides of his wife.

After the man is seated, three places are left and no other couple can sit together so the three men will have only one option left.

Hence, the number of ways in which $4$ couples be seated around a circular table such that people of same gender do not sit in adjacent positions and exactly one of the four couples sit in adjacent positions $=3!\times {}^4{C}_1\times 2=6\times 4\times 2=48$

Hence the answer is $48$.