Jonathan is leading his PHINS group across campus when he hears that Andrew got lost in the
basement of DRL. To ensure no one else gets lost in the depths of DRL, Jonathan insists that
his students travel using the “buddy” system, meaning that his students must pair up before
entering DRL. Jonathan’s group happens to consist of 864 distinct students. How many ways
can Jonathan pair up students in his group to search for Andrew? (Note that Andrew is not part
of the 864 students, the order of the students within a pair does not matter, and the ordering
of the pairs doesn’t matter either).

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.

Jonathan is leading his PHINS group across campus when he hears that Andrew got lost in the basement of DRL. To ensure no one else gets lost in the depths of DRL, Jonathan insists that his students travel using the “buddy” system, meaning that his students must pair up before entering DRL. Jonathan’s group happens to consist of 864 distinct students. How many ways can Jonathan pair up students in his group to search for Andrew? (Note that Andrew is not part of the 864 students, the order of the students within a pair does not matter, and the ordering of the pairs doesn’t matter either).

Explore the combinatorial challenge of pairing 864 students into groups using the buddy system. This problem involves advanced combinatorial concepts including factorials, permutations, and combinations. Perfect for college-level students studying discrete mathematics.

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