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?
Best Answer
Your answer is correct.
This is easy to see as follows.
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 $144$.