How to Arrange 8 Students with Specific Conditions for a Class Picture

Imagine a classroom scenario where 8 students need to be arranged in a row for a class picture. However, there are specific conditions: Paul and Peter must stand next to each other, and John and Joey also need to be together. How can we determine the number of possible arrangements while adhering to these conditions?

Method 1: Treating Pairs as Single Entities

One way to tackle this problem is to treat Paul and Peter as a single unit and John and Joey as another single unit. This reduces the problem to arranging 6 entities (the two units and the remaining 4 students) in a row.

Step 1: Arrange the two units (Paul-Peter and John-Joey) in 2! (2) ways. This accounts for the two possible orders: (Paul-Peter, John-Joey) or (John-Joey, Paul-Peter).

Step 2: Within each unit, Paul and Peter can switch places in 2! (2) ways, and John and Joey can also switch places in 2! (2) ways.

Step 3: There are 6! (720) ways to arrange the 6 entities, including the two units and the remaining 4 students.

Combining all these, the total number of ways to arrange the 8 students is calculated as:

2 (orders of units) × 2 (orders within Paul-Peter unit) × 2 (orders within John-Joey unit) × 720 (arrangements of 6 entities) 2880 ways.

Method 2: Using Factorials and Combinations

Another method involves using factorial calculations and combinations to determine the number of possible arrangements.

Step 1: Arrange the 4 remaining students, which can be done in 4! (24) ways.

Step 2: Place the two pairs in the remaining 5 spots. For the first pair, there are 5 possible positions and 2 ways to arrange them (Paul-Peter or Peter-Paul). For the second pair, there are 4 remaining positions and 2 ways to arrange them (John-Joey or Joey-John).

Combining these:

24 (arrangements of 4 students) × 5 (positions for first pair) × 2 (arrangements within first pair) × 4 (remaining positions for second pair) × 2 (arrangements within second pair) 2880 ways.

Conclusion

Using combinatorial methods, we can determine that there are 2880 possible ways to arrange the 8 students in a row for a class picture while ensuring that Paul and Peter are next to each other and John and Joey are together.

This problem demonstrates the practical application of factorial and permutation principles in real-world scenarios, such as arranging students for a big group photo session.