Six people forming three couples sit around a round table (with all the seating con-
figurations equally likely). What is the probability that, for every couple, the two people
forming the couple sit next to each other? (Such a seating arrangement would be conve-
nient if they order three large dishes, each to be shared by one couple.)

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!

Six people forming three couples sit around a round table (with all the seating configurations equally likely). What is the probability that, for every couple, the two people forming the couple sit next to each other?

Explore how to calculate the probability that three couples, sitting around a round table, sit next to each other. This problem delves into combinatorics, circular permutations, and probability, offering a detailed step-by-step solution. Suitable for high school students in Grades 10-12.

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