We are tasked with determining how many ways Jonathan can pair up his group of 864 students to search for Andrew. To solve this, we need to count the number of ways to divide the students into 432 pairs (since there are 864 students and each pair consists of 2 students).

Step 1: Total number of ways to arrange the students

First, consider how to arrange all 864 students in a line. The total number of arrangements of 864 students is given by $864!$ (864 factorial).

Step 2: Correcting for the order of students within a pair

Since the order of the students within each pair does not matter, we need to account for the fact that each pair can be arranged in 2 ways (student A first or student B first). Therefore, for each of the 432 pairs, there are 2 arrangements. This gives us $2^{432}$ as the number of different ways the pairs can be arranged within themselves.

Step 3: Correcting for the order of the pairs

Since the order of the pairs themselves does not matter, we must divide by the number of ways to arrange 432 pairs, which is $432!$.

Step 4: Formula for the number of ways to form the pairs

The total number of ways to pair up the students is:

$\frac{864!}{2^{432} \times 432!}$

This expression gives the number of ways Jonathan can pair up the 864 students.

Conclusion

The number of ways to pair up the 864 students in Jonathan's group is:

$\frac{864!}{2^{432} \times 432!}$

Would you like to see more details on how to simplify or compute such expressions?

Here are five related questions to consider:

1. What is the number of ways to pair up 6 students?
2. How would the formula change if the order of the pairs mattered?
3. What if instead of pairing, the students were grouped in triplets?
4. How can you estimate the size of large factorials like $864!$?
5. What is the combinatorial significance of pairing problems like this?

Tip: When handling large factorials, Stirling’s approximation provides a useful way to estimate their size.