In how many ways can five people sit round a circular table?

1. Analyze the Question

The question asks for the number of ways to arrange $n$ distinct objects (people) around a circular table. In circular permutations, the arrangement is considered the same if everyone shifts one position to the left or right, as the relative positions remain unchanged.

2. Explain the Concept

When arranging objects in a straight line (linear permutation), the number of ways to arrange $n$ objects is $n!$ (n factorial).

However, in a circular arrangement, we fix one person in a seat to break the symmetry. Once one person is fixed, the remaining $(n - 1)$ people can be arranged in the remaining seats in $(n - 1)!$ ways.

The formula for circular permutation is:

3. Show the Solution

• Step 1: Identify the number of people ($n$). Here, $n = 5$.
• Step 2: Apply the circular permutation formula: $$\text{Ways} = (n - 1)!$$
• Step 3: Substitute the value of $n$: $$\text{Ways} = (5 - 1)!$$ $$\text{Ways} = 4!$$
• Step 4: Calculate the factorial: $$4! = 4 \times 3 \times 2 \times 1$$ $$4 \times 3 = 12$$ $$12 \times 2 = 24$$ $$24 \times 1 = 24$$

4. Identify the Answer

The calculation gives us 24. Comparing this to the provided options, it matches option A.

5. Exam Tip

Always remember the difference between linear and circular arrangements:

• Linear: $n!$ (e.g., 5 people in a row = $5! = 120$)
• Circular: $(n - 1)!$ (e.g., 5 people at a table = $4! = 24$) A quick way to remember this is that in a circle, you "lose" one position because there is no "start" or "end" seat.