Understanding Round Table Seating Arrangements

This question asks for the number of distinct seating arrangements possible when 6 people are seated at a round table. This is a problem involving permutations, specifically circular permutations.

Circular Permutations Explained

In a linear arrangement (like a bench), if you have 6 people, there are $6!$ (6 factorial) ways to arrange them. This is because each position is distinct.

However, at a round table, arrangements are considered the same if they can be obtained by rotating the table. For example, if we have people A, B, C, D, E, F, the arrangement ABCDEF in clockwise order is the same as BCDEFA, CDEFAB, and so on, if we just rotate everyone one seat over. Since there are 6 people, there are 6 such rotations that result in the same relative arrangement.

To account for this repetition due to rotation, we fix one person's position and arrange the remaining people relative to that fixed person. This effectively turns the circular arrangement into a linear one for the remaining people.

Calculating Distinct Arrangements

The formula for the number of distinct arrangements of $n$ distinct objects in a circle is:

\(P_{circular} = (n-1)!\)

In this problem:

• The number of people, $n$, is 6.

Using the formula:

\(\text{Number of arrangements} = (6-1)!\)

$ = 5! $

Now, we calculate the factorial of 5:

\(5! = 5 \times 4 \times 3 \times 2 \times 1\)

Let's break down the calculation:

• \(5 \times 4 = 20\)
• \(20 \times 3 = 60\)
• \(60 \times 2 = 120\)
• \(120 \times 1 = 120\)

So, $5! = 120$.

Conclusion

There are 120 distinct seating arrangements possible for 6 people at a round table.