Arranging Boys and Girls Around a Circular Table: Ensuring No Two Girls Sit Together

In this article, we explore a classic combinatorial problem: determining the number of ways 10 boys and 5 girls can be seated around a circular table such that no two girls sit together. Combining the principles of permutations and combinatorics, we provide a detailed step-by-step solution, making it accessible to students and enthusiasts alike.

Introduction to the Problem

The challenge presented is a rich example of how to apply combinatorics and permutations in real-world scenarios. The problem requires us to arrange 10 boys and 5 girls in a circular fashion, ensuring that no two girls are seated adjacent to each other. This constraint introduces a unique layer of complexity, making it a perfect illustration for learners interested in understanding more advanced concepts in circular permutations and combinatorial arrangements.

Step-by-Step Solution

Step 1: Seat the Boys

The first step is to arrange the 10 boys around the circular table. Since the arrangement is circular, we choose one boy to fix his position to avoid counting rotations as different arrangements. This means there are ((10-1)! 9!) ways to arrange the remaining 9 boys.

Mathematically, this is calculated as:

(9! 362880)

Step 2: Create Slots for the Girls

Once the boys are seated, they create distinct slots where the girls can sit. With 10 boys, there are 10 slots available: one between each pair of boys and one at each end. Our goal is to strategically place the 5 girls in these slots so that no two girls are adjacent to each other.

These slots form a circle around the boys. Therefore, the number of ways to choose 5 out of these 10 slots is given by the combination formula:

(binom{10}{5} frac{10!}{5!(10-5)!} frac{10!}{5!5!} 252)

Step 3: Arrange the Girls in the Chosen Slots

After choosing the slots, we need to arrange the 5 girls in those 5 slots. The number of ways to arrange 5 girls in 5 slots is given by the factorial of 5, which is (5! 120).

Mathematically, this is calculated as:

(5! 120)

Total Arrangements

To find the total number of ways to arrange the boys and girls according to the given conditions, we multiply the number of arrangements of the boys, the number of ways to choose the slots, and the arrangements of the girls:

(Total Arrangements 9! times binom{10}{5} times 5!)

Substituting the values, we get:

(Total Arrangements 362880 times 252 times 120)

Calculating this step-by-step:

(362880 times 252 91545600)

Then,(91545600 times 120 10985472000)

Thus, the total number of ways to arrange 10 boys and 5 girls around a circular table such that no two girls sit together is:

(boxed{10985472000})

Conclusion

This detailed solution demonstrates how to solve a complex combinatorial problem using a systematic approach, combining circular permutations and combinations. The key steps involve arranging the boys, creating slots for the girls, choosing the slots, and arranging the girls in those slots. This problem not only enhances our understanding of combinatorial arrangements but also provides insights into advanced permutation techniques.