# Three married couples on a rectangular table with 6 chairs

There is a rectangular table as shown in the figure, which has three chairs on two sides each. There married couples sit on these chairs. Find the number of ways in which couples sit either in front of each other or adjacently.

What I tried:

Case 1: All couples are sitting in front of each other: $$3!\times2^3$$

$$2^3$$ is the number of ways in which the member of a couple can interchange the position and $$3!$$ is the permutation of the columns.

Case 2: One couple is sitting in front of each other and two adjacently. $$2\times 3! \times 2^3$$

$$2^3$$ is the number of ways in which the member of a couple can interchange the position and $$3!$$ is the permutations of the couples and $$2$$ is because there are two ways of arrangement as shown in the figure.

So total number of ways: $$48+96=144$$

But the given answer is $$112$$

What am I doing wrong?

• I don't see any problem with your calculation.
– lulu
May 16 at 17:15
• According to their solution, for case 2, the Number of ways=$4\times 2^4$. They haven't provided any justification. :/ May 16 at 17:19
• Since the given answer is not divisible by $3!$ it cannot be correct. May 16 at 17:21
• Permuting the three couples gives a clear factor of $3$ for each pattern.
– lulu
May 16 at 17:21
• $4\times 2^4$ is weird ! May 16 at 17:29

There are $$3$$ fundamentally different arrangements.
The three couples can be permuted in $$3!$$ ways.
The partners can be permuted in $$2^3$$ ways.
The product of these is $$144$$.