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.

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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.