Teddy, Becky, and Jose are going to sit on a bench to wait for the bus. How many different arrangements can they sit on the bench so that Becky and Jose are sitting next to each other?
To solve this problem, we can consider Becky and Jose as a single entity. Let's call this entity BJ.
Now, we have three entities: Teddy, BJ, and the empty seat. We can arrange these three entities in three different ways:
1. Teddy - BJ - (Empty seat)
2. BJ - Teddy - (Empty seat)
3. (Empty seat) - BJ - Teddy
However, within the BJ entity, Becky and Jose can switch positions. Thus, we need to consider the possible arrangements within the BJ entity.
Becky and Jose can be arranged in two ways:
1. Becky - Jose
2. Jose - Becky
Therefore, the total number of different arrangements is 3 (possible arrangements of the entities) multiplied by 2 (possible arrangements within the BJ entity).
So, there are 3 x 2 = <<3*2=6>>6 different arrangements.
To calculate the number of different arrangements, we need to consider the possible positions for Becky and Jose to sit together.
1. We can treat Becky and Jose as a single unit. So, consider Teddy and "Becky&Jose" as two entities.
2. We can arrange these two entities and Teddy on the bench in different ways.
Let's go step-by-step to find the number of different arrangements:
Step 1: Treat Becky and Jose as a single unit ("Becky&Jose").
Step 2: Arrange Teddy, "Becky&Jose," and the remaining empty seat(s) on the bench.
For each arrangement, Becky and Jose will always be seated next to each other.
So, the number of different arrangements is equal to the number of arrangements of Teddy, "Becky&Jose," and the empty seat(s).
Step 3: Calculate the number of different arrangements of Teddy, "Becky&Jose," and the empty seat(s).
Since we have 3 individuals (Teddy, "Becky&Jose"), there are only 2 possible ways to arrange them:
1. Teddy + "Becky&Jose"
2. "Becky&Jose" + Teddy
Step 4: Calculate the number of different arrangements for each case.
Case 1: Teddy + "Becky&Jose"
The arrangement would be "Teddy, Becky, and Jose" or "Teddy, Jose, and Becky."
Case 2: "Becky&Jose" + Teddy
The arrangement would be "Becky, Jose, and Teddy."
Step 5: Add the number of different arrangements from each case.
Case 1: 2 arrangements
Case 2: 1 arrangement
Total number of different arrangements = 2 + 1 = 3
Therefore, there are 3 different arrangements in which Becky and Jose are sitting next to each other on the bench.