Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)
1. **Fix the position of person with badge 2:**
   - Since the table is circular, we can fix one person to eliminate rotational symmetry. Let's fix person 2 at a specific position.

2. **Determine possible arrangements:**
   - We need to arrange the remaining persons (1, 3, 4, 5) such that no two persons with consecutive numbers are seated next to each other.
   - Let's consider the fixed position of person 2. We can place person 2 at any of the 5 positions around the table. 

3. **Identify valid sequences:**
   - We need to find sequences where no two consecutive numbers are adjacent. Let's list the valid sequences:
     - If person 2 is fixed, the valid sequences are:
       - 2, 4, 1, 3, 5
       - 2, 5, 3, 1, 4
   - These are the only two valid sequences that satisfy the condition.

4. **Count the total number of arrangements:**
   - Since person 2 can be fixed in any of the 5 positions, and for each fixed position, there are 2 valid sequences, the total number of arrangements is:
     \[
     5 \times 2 = 10
     \]

The final answer is \(\boxed{10}\).