Sure, here is a test case generator for the given problem:

```python
import random

def tcgen():
    # Generate N and K such that 1 ≤ K ≤ N ≤ 100
    N = random.randint(1, 100)
    K = random.randint(1, N)
    
    return N, K

# Example usage
if __name__ == "__main__":
    test_case = tcgen()
    print(test_case)
```

This function will generate random values for \(N\) and \(K\) within the specified constraints. \(N\) is the total number of balls, and \(K\) is the number of persons, ensuring 1 ≤ \(K\) ≤ \(N\) ≤ 100.

Here is a sample output of the test case generator:

```
(75, 25)
```

Explanation:
- \(N = 75\): There are 75 balls to distribute.
- \(K = 25\): There are 25 persons to distribute the balls to.

You can run this function multiple times to generate different test cases for the problem.