In how many ways can 5 people sit around a circular table

Question

In how many ways can 5 people be seated around a circular table?

Solution — Step by Step

In a linear arrangement of 5 people (in a row), every position is distinct — leftmost, second, etc. So there are $5! = 120$ arrangements.

In a circular arrangement, there is no “first position.” The table looks the same if everyone shifts one seat clockwise. Two circular arrangements are considered identical if one can be obtained from the other by rotation.

To count distinct circular arrangements, fix one person in a seat (call them Person A). This eliminates the redundancy from rotations.

Now the remaining 4 people can be arranged in the remaining 4 seats in any order.

With Person A fixed, the remaining 4 people can be arranged in:

4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways}

So the number of distinct circular arrangements = 24.

The general formula for circular permutations of $n$ distinct objects:

\text{Circular arrangements} = (n-1)!

For $n = 5$ : $(5-1)! = 4! = 24$ ✓

Why This Works

The formula $(n-1)!$ comes from the “fix one” argument. In linear permutations, $n!$ counts all arrangements, but each distinct circular arrangement is counted $n$ times (once for each rotation). So:

\text{Distinct circular arrangements} = \frac{n!}{n} = (n-1)!

The divisor $n$ represents the $n$ rotational equivalences we’ve collapsed.

Alternative Method — Direct Reasoning

Think of it as assigning seats. Seat Person A anywhere (1 way, since the table is symmetric). Then:

Person B: 4 remaining seats → 4 choices
Person C: 3 remaining seats → 3 choices
Person D: 2 remaining seats → 2 choices
Person E: 1 remaining seat → 1 choice

Total: $4 \times 3 \times 2 \times 1 = 4! = 24$ .

If the problem also asks about a necklace (where the arrangement can be flipped — clock-wise and anticlockwise are the same), divide by 2 more: $(n-1)!/2 = 4!/2 = 12$ . For a round table, usually clockwise and anticlockwise are different (you can tell by looking at who is to your left vs right). Always check the problem statement.

Common Mistake

The classic error: writing $5! = 120$ without accounting for the circular nature. In a circular arrangement, the 120 linear arrangements collapse into $120/5 = 24$ distinct ones. Many students forget to divide by $n$ (or equivalently, forget to “fix” one person). For any circular arrangement question, immediately write $(n-1)!$ , not $n!$ .

Question

Solution — Step by Step

Why This Works

Alternative Method — Direct Reasoning

Common Mistake

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