For a given positive integer m, you have a round table of 2m seats, and m couples attending this party (all 2m people are distinct). Given an odd integer 1 < n < m, you want to seat everyone such that the two people in each couple are exactly n seats apart (as in there are n−1 seats between them). How many distinct ways are there to achieve this? (Note: Rotations and reflections of an arrangement are considered distinct.)

To solve this problem, let's break it down step by step.

### Step 1: Understand the seating arrangement

We have \( m \) couples and \( 2m \) seats around a circular table. The problem specifies that for a given odd integer \( 1 < n < m \), the two members of each couple must be seated exactly \( n \) seats apart.

### Step 2: Consider the seating of one couple

Consider the first couple. Let's fix one member of this couple, say \( A_1 \), at seat 1. The second member, \( B_1 \), must be seated exactly \( n \) seats apart. Since we have a circular arrangement, the possible positions for \( B_1 \) are either seat \( 1+n \) or \( 1-n \) modulo \( 2m \). However, since \( n \) is odd, the position \( 1-n \) modulo \( 2m \) is the same as \( 1+n \) modulo \( 2m \) when considering only distances, so we only consider one valid position for \( B_1 \).

### Step 3: Calculate permutations for other couples

Now, we have seated \( A_1 \) and \( B_1 \). This leaves \( 2m-2 \) seats and \( m-1 \) couples. The next step is to place the remaining \( m-1 \) couples. For each subsequent couple, after fixing the position of one partner, there is exactly one valid position for the other partner that is \( n \) seats apart.

### Step 4: Consider distinct arrangements

Since the seating is circular, normally we would consider rotations equivalent. However, the problem specifically states that rotations and reflections are distinct, so no symmetry reductions are needed. 

Given that for each couple there is only one way to position the second member, the number of distinct arrangements of the couples is simply the number of permutations of the couples. There are \( m! \) ways to arrange \( m \) couples.

### Step 5: Conclusion

Thus, the total number of distinct ways to seat the \( 2m \) people so that each couple is \( n \) seats apart is \( \boxed{m!} \).

Explore how to count distinct ways to seat 2m people around a circular table such that each of m couples is n seats apart. Learn about circular arrangements and permutations in combinatorics. Suitable for advanced mathematics enthusiasts.

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