To find the number of different ways for Dan, Ben, and Mark to sit together, we can treat them as one entity and calculate the number of arrangements with this entity and the other two classmates seated in the row.
Step 1: Treat Dan, Ben, and Mark as one entity.
Since Dan, Ben, and Mark want to sit together, we can treat them as one group. Let's call this group "DBM."
Step 2: Arrange the group "DBM" and the two classmates.
Now, we have four entities to arrange: DBM, classmate 1, classmate 2, and classmate 3.
To calculate the number of ways to arrange these entities, we can think of arranging them as filling in the seats from left to right.
- First, arrange the group "DBM" as one entity. There is only one way to arrange them since they want to sit together.
- Next, arrange the two classmates. There are two classmates remaining to be seated, so we have two options for the first classmate and one option for the second classmate.
Therefore, the total number of ways to arrange DBM and the two classmates is 1 * 2 * 1 = 2.
Step 3: Consider the arrangements within the group "DBM."
Within the group "DBM," Dan, Ben, and Mark can arrange themselves in 3! = 3 * 2 * 1 = 6 different ways.
Step 4: Combine the results.
To get the final answer, we multiply the number of arrangements from Step 2 by the number of arrangements within DBM from Step 3:
Total Number of Ways = Number of arrangements within DBM * Number of arrangements of DBM with classmates
= 6 * 2
= 12.
Therefore, there are 12 different ways for everyone to sit if Dan, Ben, and Mark want to sit together.