there are 4 students sitting on the stage at the school assembly waiting to receive their prizes.Alan and Julia wanted to sit together. If they did,how many different arrangements of students would there be?
consider Alan and Julia as one person.
So, there are really only 3 students, making 3! = 6 arrangements.
However, Alan and Julia may switch places, so there are 12 total arrangements.
Sitting in the school cafeteria you notice the different groups of students sitting together. Which group would you think might be characterized as a clique?
To determine the different arrangements of students, we need to consider the seating arrangement requirement that Alan and Julia should sit together.
First, let's think about Alan and Julia as a single entity. So, we have 3 entities - Alan & Julia (considered as one), and the other two students.
Now, to calculate the different arrangements, we can treat Alan & Julia as a single item that can be arranged with the other two students. This can be done in 2 ways: Alan & Julia together on the left, or Alan & Julia together on the right.
For each arrangement, the remaining two students can be arranged in 2! = 2 factorial ways, which is 2.
Therefore, the total number of different arrangements with Alan and Julia sitting together would be 2 x 2 = 4.
So, there would be 4 different arrangements of the students if Alan and Julia sat together on the stage.