Step 1: Understanding the Concept:
This problem requires us to find arrangements in a circle with a restriction ("no two girls sit together"). To enforce this separation, we first seat the unrestricted group (boys) around the table. This creates spaces or "gaps" between them. We then place the restricted group (girls) into these gaps to ensure they are separated by at least one boy.
Step 2: Key Formula or Approach:
- The number of ways to arrange $n$ distinct objects in a circle is $(n-1)!$.
- Once objects are seated in a circle, the spaces between them become distinct linear positions. The number of ways to arrange $r$ objects in $n$ distinct available positions is ${}^n P_r = \frac{n!}{(n-r)!}$.
- The total number of arrangements is the product of the two independent steps.
Step 3: Detailed Explanation:
First, we arrange the 6 boys around the round table.
The number of ways to arrange 6 boys in a circle is:
\[ (6 - 1)! = 5! = 120 \]
Once the 6 boys are seated, they create exactly 6 gaps between them around the table. Since the boys are distinct individuals and are already fixed in their positions, these 6 gaps are distinguishable from each other.
We have 5 girls to seat, and they must not sit together. Thus, we must place at most one girl in each of the 6 available gaps.
The number of ways to seat the 5 distinct girls into the 6 distinct gaps is given by permutations:
\[ {}^6 P_5 = \frac{6!}{(6-5)!} = \frac{6!}{1!} = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]
The total number of valid seating arrangements is the product of the number of ways to seat the boys and the number of ways to seat the girls:
\[ \text{Total ways} = 120 \times 720 \]
\[ \text{Total ways} = 12 \times 72 \times 100 = 864 \times 100 = 86400 \]
Step 4: Final Answer:
The total number of ways is 86400.