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.)
Thank you
To determine the number of ways the 4 boys and 4 girls can sit around a circle table with all the boys sitting together, we can treat the group of 4 boys as a single entity. Let's call them the "boys group."
Now, we have essentially 5 entities to arrange around the table: the boys group and the 4 individual girls.
To arrange the 5 entities around the circle, we need to consider the number of ways the boys group can be arranged within itself, as well as the number of ways the boys group and the girls can be arranged as a whole.
Let's start with arranging the boys within themselves:
Since the boys group needs to sit together, we can consider them as a single entity. Thus, the boys group can be arranged among themselves in 4! (4 factorial) ways.
Next, let's consider the boys group and girls as a whole:
Now, we have 5 entities to arrange around the table - the 4 individual girls and the boys group. We can arrange them in (5-1)! = 4! ways.
Lastly, we need to consider the rotation of the same arrangement:
Since the table is circular, we need to account for rotations. But in this case, since the boys group is considered as a single entity, we don't need to consider different rotations of the boys within themselves. So, we only need to consider the rotation of the entire arrangement.
For any arrangement, there are 8 different starting positions around the table. However, since rotations of the same arrangement are considered the same, we divide the total number of arrangements by 8.
Putting it all together, the total number of ways the 4 boys and 4 girls can sit around a circle table with all the boys sitting together is:
(4! * 4!) / 8
Calculating the expression, we get:
(24 * 24) / 8 = 72
Therefore, there are 72 different ways the boys and girls can sit around the table with all the boys sitting together.