In how many ways can 4 boys and 4 girls sit around a circle table if each boy sits directly between two girls? (Rotations of the same arrangement are still considered the same. Each boy and girl is unique, not interchangeable.)
To find the number of ways the boys and girls can sit around the circle table, satisfying the given condition, we can use the concept of permutations.
First, let's fix one girl at the top of the table. We have 4 choices for this girl.
Next, let's consider the positioning of the boys. Since each boy sits directly between two girls, there are two potential positions for each boy. This means we have 2 options for each boy, and since there are 4 boys, there are 2^4 = 16 possible arrangements of the boys.
Now, we have to factor in the rotation. Since rotations of the same arrangement are considered the same, we need to divide the total number of arrangements by the number of possible rotations. In this case, there are 8 possible rotations since there are 8 people sitting at the table.
Finally, to get the total number of ways, we multiply the number of choices for the fixed girl by the number of possible arrangements of the boys and divide by the number of rotations. Therefore, the total number of ways is (4 * 16) / 8 = 8.
Hence, there are 8 different ways the boys and girls can sit around the circle table, with each boy sitting directly between two girls.