Dan, Ben, and Mark are friends. Together, along with two of their classmates, they bought movie tickets that were all in one row. How many different ways can everybody sit if Dan, Ben, and Mark want to sit together?
There are 5*4*3*2*1 = 5! = 120 ways total to have them seated. Call every person A, B, C, D, and E.
Set A, B, and C as Dan, Ben, and Mark, and the two other people as D and E. Make A, B, and C variable X.
The number of ways of X, D, and E sitting together now is 3*2*1 = 3! = 6.
But don't forget, there are 3*2*1 = 3! = 6 ways of making X:
1. ABC
2. ACB
3. CBA
4. CAB
5. BAC
6. BCA
So, the final answer is 6*6 = 6^2 = 36 different ways for everyone to sit if Dan, Ben, and Mark want to sit together.
Why did the friends go to see a movie about chairs? Because they wanted to experience the "seating" arrangement! If Dan, Ben, and Mark want to sit together, we can treat them as a single unit. So we have three friends considered as one, plus two classmates. Now we have 4 units to arrange in a row. The number of ways to do this is 4! (four-factorial), which equals 4 x 3 x 2 x 1 = 24. So, there are 24 different ways everybody can sit if Dan, Ben, and Mark want to be together. That's a lot of seat-arranging possibilities!
To find the number of different ways that Dan, Ben, and Mark can sit together, we can consider them as a single unit. Let's call this unit "DBM".
Now, we have 3 units: DBM, Classmate 1, and Classmate 2. We can arrange these 3 units in 3! (3 factorial) ways.
Within the DBM unit, Dan, Ben, and Mark can arrange themselves in 3! ways.
So, the total number of different ways that everybody can sit is 3! * 3!.
Calculating this expression: 3! * 3! = 6 * 6 = 36.
Therefore, there are 36 different ways that everybody can sit if Dan, Ben, and Mark want to sit together.
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.