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.)
Incorrect, It is 144
To solve this problem, we need to find the number of seating arrangements such that all the boys sit together. We can treat all the boys as a single entity or group.
1. Consider the group of boys as a single entity: Now we have 5 objects (boy group and 4 girls) to arrange around a circular table. The number of ways to arrange these objects in a circle is (n-1)!, where n is the number of objects. Therefore, we have (5-1)! = 4! = 24 ways to arrange the objects around the table.
2. Within the boy group: Now, within the boy group, we need to arrange the 4 boys. The number of ways to arrange these 4 boys is 4!.
3. Therefore, the total number of seating arrangements where all the boys sit together is 4! × 4! = 24 × 24 = 576.
Hence, there are 576 ways for all the boys to sit together around a circular table with 4 girls.
Visualize them all in a row.
The four boys can be arranged in 4x3x2x1
and then the girls can be arranged in 4x3x2x1 ways
which is 24x24 or 576 ways
Now if we seat them there are 8 possible rotations without changing the order, so
number of ways to seat them is 576/2 or 288 ways