Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?
1. **Fixing the Coaches**: Since the two coaches must sit opposite each other, we can fix one coach in a position. There are 2 ways to place the second coach opposite the first one. This is because once we place the first coach, the second coach has only one position available directly opposite.

2. **Arranging the Mathletes**: With the positions of the coaches fixed, we now have 4 remaining seats for the 4 mathletes. The number of ways to arrange 4 distinct mathletes in 4 seats is given by the factorial of 4, which is $4!$.

   \[
   4! = 4 \times 3 \times 2 \times 1 = 24
   \]

3. **Combining the Arrangements**: Since there are 2 ways to place the coaches and 24 ways to arrange the mathletes, the total number of distinct arrangements is:

   \[
   2 \times 24 = 48
   \]

However, since the table is circular, we need to account for rotational symmetry. In a circular arrangement, rotating the entire arrangement does not create a new distinct arrangement. For 6 people, there are 6 possible rotations, but since the coaches are fixed opposite each other, there are only 2 unique rotations (one for each coach's position). Therefore, we divide by 2 to account for this symmetry:

   \[
   \frac{48}{2} = 24
   \]

Thus, the total number of distinct arrangements is:

   \[
   \boxed{24}
   \]