In how many ways can 6 people sit around a table

1. Analyze the question

The question asks for the number of ways to arrange 6 people around a table. In mathematics, this is a problem of circular permutation. Unlike linear arrangements (where there is a clear start and end), circular arrangements are relative to each other.

2. Explain the concept

When arranging $n$ distinct objects in a straight line, the number of ways is $n!$ (n factorial). However, when arranging $n$ objects in a circle, rotating the entire arrangement does not create a new unique arrangement.

To account for this, we fix one person in a seat to serve as a reference point. Once that 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, $n = 6$.
• Step 2: Apply the circular permutation formula. $$\text{Ways} = (6 - 1)!$$
• Step 3: Calculate the factorial. $$\text{Ways} = 5!$$
• Step 4: Expand the factorial. $$5! = 5 \times 4 \times 3 \times 2 \times 1$$ $$5 \times 4 = 20$$ $$20 \times 3 = 60$$ $$60 \times 2 = 120$$ $$120 \times 1 = 120$$

4. Identify the answer

The calculation results in 120. Comparing this to the provided options, it matches option A.

5. Tip

Always remember: "Linear is $n!$, Circular is $(n-1)!$." If the question mentions a "round table" or "circular arrangement," immediately subtract 1 from the total number of items before calculating the factorial. This simple trick will save you time in the exam hall!