To solve this problem, we need to arrange 6 boys and 3 girls such that:

1. No two girls sit together.
2. No two boys sit together.

The solution requires strategic placement of boys and girls.

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Step 1: Arranging the boys

We start by arranging the 6 boys. Since they are all distinct, the total number of ways to arrange them is:

$6! = 720$

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Step 2: Placing the girls

To ensure that no two boys or girls sit together, we need to alternate the boys and girls. This requires placing the girls in the spaces between the boys. For example:

$B \, G \, B \, G \, B \, G \, B \, B \, B$

There are 7 slots around and between the boys (before, after, and in between). We need to select 3 of these 7 slots for