1. The Total Number of Outcomes

The number of ways to choose 8 people out of 14 is given by the combination formula $\(\binom{n}{k}\)$ :

$$
\(\binom{14}{8}=\frac{14!}{8!6!}=3003\)
$$

2. The Number of Successful Outcomes

We need to select exactly $\(\mathbf{3}\)$ couples and then $\(\mathbf{2}\)$ individuals who are the same gender but not part of a 4th couple.

Step A: Choose the $\mathbf{3}$ couples There are $\(\mathbf{7}\)$ couples available. We choose 3:

$$
\(\binom{7}{3}=35 \text { ways }\)
$$


Step B: Choose the remaining $\mathbf{2}$ people (Same Gender) We have $\(\mathbf{7}-\mathbf{3}=\mathbf{4}\)$ couples remaining ( 8 people: 4 men and 4 women). To ensure we don't accidentally pick a 4th couple, we must pick our 2 people from different remaining couples.
- Case 1: Two Men. We choose 2 couples out of the remaining 4, and take the man from each:

$$
\(\binom{4}{2}=6 \text { ways }\)
$$

- Case 2: Two Women. We choose 2 couples out of the remaining 4, and take the woman from each:

$$
\(\binom{4}{2}=6 \text { ways }\)
$$

Total ways to pick the 2 extra people: $6+6=12$ ways.
Step C: Calculate total successful combinations Multiply the choices together:

$$
\(35(\text { couples }) \times 12(\text { individuals })=420 \text { ways }\)
$$

3. Final Probability Calculation

The probability ( $P$ ) is the successful outcomes over the total outcomes:

$$
\(P=\frac{420}{3003}\)
$$


We can simplify this by dividing both numerator and denominator by 21 :

$$
\(P=\frac{20}{143}\)
$$