In how many ways can a group of four girls and four boys seated at a

round table if boys and girls must alternate?

In a problem like this, seating is considered in a **circular arrangement**, so we need to take that into account, along with the condition that boys and girls must alternate. Here’s the step-by-step breakdown:

### Step 1: Fix one person to eliminate rotational symmetry
In a circular seating arrangement, rotations of the same arrangement are considered equivalent. Therefore, we can fix the position of one person (say, one of the boys) to avoid counting identical rotations as distinct. This essentially converts the circular arrangement into a linear arrangement for counting purposes.

### Step 2: Arrange the remaining people
- After fixing one boy in a specific seat, there are \(3!\) ways to arrange the other three boys in the remaining seats designated for boys (since we fixed one boy).
- Similarly, there are \(4!\) ways to arrange the four girls in the seats designated for girls (since their positions also alternate).

### Step 3: Multiply the two results
The total number of ways to arrange the group is the product of the number of ways to arrange the boys and the number of ways to arrange the girls.

\[
\text{Total number of ways} = 3! \times 4! = 6 \times 24 = 144
\]

Thus, the group of four boys and four girls can be seated at the table in **144 ways**.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What would happen if the number of boys and girls were unequal?
2. How would the solution change if it was a linear arrangement, not circular?
3. What if boys and girls do not have to alternate?
4. How many ways can you arrange them if two specific girls must sit next to each other?
5. What would the result be if there were 5 boys and 5 girls?

**Tip**: In circular permutations, always account for rotational symmetry to avoid overcounting!

In how many ways can a group of four girls and four boys be seated at a round table if boys and girls must alternate?

Learn how to determine the number of ways to seat 4 boys and 4 girls alternately at a round table. This problem explores circular permutations and factorials, providing a step-by-step solution. Suitable for high school students in Grades 9-12.

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