Explanation:

- First, we should calculate the volume of the three initial spheres:

- Volume of sphere with radius 2 cm: \( \frac{4}{3} \pi (2)^3 = \frac{32}{3} \pi \).

- Volume of sphere with radius 4 cm: \( \frac{4}{3} \pi (4)^3 = \frac{256}{3} \pi \).

- Volume of sphere with radius 6 cm: \( \frac{4}{3} \pi (6)^3 = \frac{864}{3} \pi \).

- Add these volumes together:

- Total initial volume = \( \frac{32}{3} \pi + \frac{256}{3} \pi + \frac{864}{3} \pi = \frac{1152}{3} \pi \).

- There is a 25% material loss, so:

- Effective volume = \( 0.75 \times \frac{1152}{3} \pi = 288 \pi \).

- Use the volume formula for spheres to find the radius \( r \) of the new sphere:

- \( \frac{4}{3} \pi r^3 = 288 \pi \).

- Solve for \( r \): \( r^3 = 216 \), so \( r = 6 \).

- Option 1: \( \mathbf{6 \, \text{cm}} \) is the correct choice.