Sure, I can help you generate test cases for the given problem. The problem seems to involve combinatorial calculations, and the constraints are fairly large. Here's a test case generator:

```python
import random

def tcgen():
    # Constraints
    MAX_N = 3000
    MAX_M = 3000

    # Randomly generate N and M within their respective ranges
    N = random.randint(1, MAX_N)
    M = random.randint(1, MAX_M)

    return N, M

# Generate a test case
N, M = tcgen()
print(N, M)
```

This generator will produce random values for \( N \) and \( M \) within their respective limits, ensuring that the test cases conform to the problem's constraints. You can run this generator multiple times to create multiple test cases for the problem.

Here’s an example of running the generator and printing the output:

```python
for _ in range(10):  # Generate 10 test cases
    N, M = tcgen()
    print(f"{N} {M}")
```

This will print 10 different test cases, each with random values for \( N \) and \( M \) within the specified ranges.