In how many ways can 3 boys and 3 girls be seated alternately in a round table?

To solve the problem of seating 3 boys and 3 girls alternately around a round table, we can follow these steps:

### Step 1: Fix one person's position
In circular permutations, to avoid equivalent rotations being counted multiple times, we usually fix one person's position. Let's fix one boy's position. Now, we need to arrange the remaining 2 boys and 3 girls alternately around the table.

### Step 2: Arrange the remaining boys
Since one boy's position is fixed, there are \(2!\) ways to arrange the remaining 2 boys.

### Step 3: Arrange the girls
The 3 girls can be arranged in the 3 remaining positions. There are \(3!\) ways to arrange the girls.

### Step 4: Calculate the total number of arrangements
Multiply the number of ways to arrange the boys and the girls.

\[
\text{Total arrangements} = 2! \times 3! = 2 \times 6 = 12
\]

So, there are **12** different ways to seat 3 boys and 3 girls alternately around a round table.

---

Would you like more details or have any questions about the solution? Here are some related questions:

1. What if the table was linear instead of round?
2. How would the answer change if there were 4 boys and 4 girls?
3. What if the seating didn't have to be alternate?
4. How many ways can 3 boys and 2 girls be seated alternately at a round table?
5. What if the seating arrangement must start with a girl?

**Tip:** When dealing with circular permutations, remember to fix one position to simplify counting the unique arrangements.

Explore how to calculate the number of ways to seat 3 boys and 3 girls alternately around a round table using permutations. This problem involves circular permutations and factorial calculations. Suitable for high school students studying permutations and combinations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation