We are given a problem where six people, forming three couples, sit around a round table. We are tasked with finding the probability that for every couple, the two people forming the couple sit next to each other.

Let's approach the problem step-by-step:

Step 1: Total number of seating arrangements (without restriction)

Since the table is round, we can fix one person to avoid counting identical rotations multiple times. For the remaining 5 people, we can arrange them in any order. Therefore, the number of ways to arrange 6 people around a round table is equivalent to arranging 5 people in a line:

$\text{Total seating arrangements} = 5! = 120$

Step 2: Counting favorable outcomes (couples sit together)

We now want to count how many seating arrangements satisfy the condition that every couple sits together.

Each couple can be treated as a "block" of two people. Since there are three couples, we can treat them as 3 blocks.

• There are $3!$ ways to arrange the three blocks around the round table. Again, because the table is round, we fix one block and arrange the other two, which gives:

$\text{Arrangements of blocks} = 2! = 2$

• Within each couple, there are 2 possible ways to arrange the two people (one person on the left or the right). Since there are 3 couples, we have $2^3 = 8$ ways to arrange the people within each block.

Step 3: Total number of favorable outcomes

Thus, the total number of favorable seating arrangements is:

$\text{Favorable seating arrangements} = 2! \times 8 = 2 \times 8 = 16$

Step 4: Probability

The probability is the ratio of favorable outcomes to the total number of seating arrangements. Therefore, the probability that each couple sits next to each other is:

$\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{16}{120} = \frac{2}{15}$

Thus, the probability is:

$\boxed{\frac{2}{15}}$

Do you want further clarification or more details on any step?

Here are 5 related questions that can expand your understanding:

1. What would be the probability if there were 4 couples instead of 3?
2. How would the solution change if the seating were in a straight line instead of a round table?
3. What if only one couple is required to sit together—how would you calculate the probability?
4. How does the approach differ if the individuals within a couple are distinguishable versus indistinguishable?
5. Can this problem be extended to more complex arrangements, like n people forming m couples?

Tip: When working with circular permutations, always remember to fix one position to avoid overcounting rotationally identical arrangements!