An interest group of 8 members consists of one married couple together with 4 other men and 2 other women. Find the number of ways that the 8 people can sit at a round table if An executive committee consisting of a Chairman, Vice-Chairman and Secretary is to be selected from the group of 8 members.

1. Group the married couple as a single unit and find the number of circular permutations for the units.

Treating the married couple as $1$ unit, there are $6$ other people, giving $1 + 6 = 7$ units in total to arrange.
Number of ways to arrange $7$ units in a circle $= (7 - 1)! = 6! = 720$

2. Account for the internal arrangement of the married couple.

The married couple can swap seats with each other in $2! = 2$ ways.
Total number of ways $= 720 \times 2 = 1440$

1. Find the number of relative circular arrangements of the 5 men.

There are a total of $1 + 4 = 5$ men and $1 + 2 = 3$ women.
First, consider the arrangements around the round table as unnumbered (finding the relative positions).
Number of ways to arrange the 5 men in a circle $= (5 - 1)! = 24$.

2. Find the number of ways to place the 3 women such that they are separated.

The 5 men create 5 spaces between them. To ensure no two women sit together, we must slot the 3 women into 3 of these 5 spaces.
Number of ways to arrange the 3 women $= {^5P_3} = 5 \times 4 \times 3 = 60$.
Total unnumbered relative circular arrangements $= 24 \times 60 = 1440$.

3. Account for the numbered seats.

Since the 8 seats at the round table are numbered, the exact seat assignments matter (which breaks the rotational symmetry). Each valid relative circular arrangement can be rotated to start at any of the 8 distinct numbered seats.
Total number of ways $= 1440 \times 8 = 11520$.

1. Find the total number of possible committees without restrictions.

The group consists of $8$ members in total.
Since there are $3$ distinct roles (Chairman, Vice-Chairman, Secretary), order matters.
Total possible ways $= ^8P_3 = 8 \times 7 \times 6 = 336$

2. Find the number of ways to form the committee such that both members of the married couple are included.

If both members of the married couple are included, we need $1$ more member from the remaining $6$ people to form a committee of $3$.
Number of ways to choose the third member $= ^6C_1 = 6$
These $3$ selected members can be assigned to the $3$ distinct roles in $3!$ ways.
Number of ways both are included $= 6 \times 3! = 36$

3. Subtract the number of committees with both members from the total to find the required number of ways.

Number of ways with at most one member of the couple included $= 336 - 36 = 300$