In how many ways can 4 boys and 4 girls sit around a circle table if all the boys sit together? (Rotations of the same arrangement are still considered the same. Each boy and girl is unique, not interchangeable.)
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The arrangement around the table must be of the form BGBGBGBG. We can place the first boy anywhere; we then have $3! = 6$ ways of placing the remaining boys and $4! = 24$ ways of placing the girls. This gives a total of $6\cdot 24 = \boxed{144}$ arrangements
To solve this problem, we can think of the group of 4 boys as a single entity. So, we will have 5 entities in total - the group of 4 boys and the 4 girls.
Now, let's fix the position of the group of 4 boys. Since they need to sit together, we can think of them as a single unit. This gives us 5 units to arrange around the circular table.
The number of ways to arrange these entities around the circle is given by the formula (n-1)!, where n is the number of entities. In this case, we have 5 entities to arrange, so the number of ways is (5-1)! = 4!.
However, we need to consider that within the group of 4 boys, they can be arranged among themselves in 4! ways. So, we need to multiply the previous result by 4!.
Therefore, the total number of ways for the 4 boys and 4 girls to sit around the table, with all the boys sitting together, is 4! * 4! = 24 * 24 = 576 ways.