Key Concepts and Formulas

• Circular Permutations: The number of ways to arrange $n$ distinct objects around a circle is $(n-1)!$.
• Permutation of Distinct Objects: The number of ways to arrange $n$ distinct objects in a line is $n!$.
• Complementary Counting: The number of ways an event A occurs is the total number of ways minus the number of ways event A does not occur.

Step-by-Step Solution

Step 1: Calculate the Total Number of Arrangements Without Restrictions

We first determine the total number of ways to seat 5 boys and 3 girls around a circular table without any restrictions. This will serve as our base for using complementary counting.

• Total number of people: We have 5 boys + 3 girls = 8 people.
• Apply the circular permutation formula: With 8 people around a circular table, the total number of arrangements is $(8-1)! = 7!$.

Thus, the total number of arrangements is: $\text{Total arrangements} = (8-1)! = 7! = 5040$

Step 2: Calculate the Number of Arrangements Where $B_1$ and $G_1$ Sit Together

Next, we calculate the number of arrangements where the particular boy $B_1$ and the particular girl $G_1$ sit next to each other.

• Treat $B_1$ and $G_1$ as a single unit: Consider $B_1$ and $G_1$ as a single entity.
• Determine the number of entities to arrange: We now have the combined unit ($B_1G_1$), 4 remaining boys, and 2 remaining girls. This gives us a total of 1 + 4 + 2 = 7 entities to arrange around the table.
• Arrange these 7 entities circularly: The number of ways to arrange these 7 entities around the circular table is $(7-1)! = 6!$.
• Consider the internal arrangement of $B_1$ and $G_1$: Within the combined unit, $B_1$ can be to the left of $G_1$ or $G_1$ can be to the left of $B_1$. This gives us 2! = 2 ways to arrange them internally.

Therefore, the number of arrangements where $B_1$ and $G_1$ sit together is: $\text{Arrangements with } B_1 \text{ and } G_1 \text{ together} = 6! \times 2! = 720 \times 2 = 1440$

Step 3: Calculate the Number of Arrangements Where $B_1$ and $G_1$ Do Not Sit Together

Using complementary counting, we subtract the number of arrangements where $B_1$ and $G_1$ sit together from the total number of arrangements to find the number of arrangements where they do not sit together.

$\text{Arrangements with } B_1 \text{ and } G_1 \text{ not together} = \text{Total arrangements} - \text{Arrangements with } B_1 \text{ and } G_1 \text{ together}$ $= 7! - (6! \times 2!)$ We can factor out $6!$ to simplify: $= (7 \times 6!) - (2 \times 6!)$ $= (7 - 2) \times 6!$ $= 5 \times 6!$ Calculating the numerical value: $= 5 \times 720 = 3600$

Final Answer

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy $B_1$ and a particular girl $G_1$ never sit adjacent to each other is $5 \times 6!$. This corresponds to option (A).

Common Mistakes & Tips

• Forgetting Internal Arrangements: When treating individuals as a single unit, remember to account for the internal arrangements within that unit.
• Circular vs. Linear Permutations: Always use $(n-1)!$ for circular arrangements of $n$ distinct objects.
• Applying Complementary Counting Correctly: Ensure you subtract the arrangements where the unwanted condition occurs from the total arrangements.

Summary

This problem showcases how complementary counting can simplify complex permutation problems. By finding the total number of arrangements and subtracting the arrangements where the unwanted condition ($B_1$ and $G_1$ sitting together) occurs, we efficiently arrive at the solution. The final answer is $5 \times 6!$.

The final answer is \boxed{5 \times 6!} which corresponds to option (A).