Deflection is an important aspect to consider in the design of structural elements like beams. It refers to the bending or sagging of a beam under the applied load. In this case, we are given a simply supported RCC beam carrying a uniform distributed load and we need to determine the deflection at the center when both ends of the beam are fixed.

1. Understanding the Problem:
- Simply Supported RCC Beam: It means that the beam is supported on both ends and can freely rotate at these points.
- Uniform Distributed Load: It is a load that is evenly distributed along the length of the beam.
- Deflection: It is the vertical displacement or sagging of the beam under the applied load.

2. Simply Supported Beam with Uniform Distributed Load:
- When a simply supported beam carries a uniform distributed load, the maximum deflection occurs at the center of the beam.
- The deflection at the center can be calculated using the formula: δ = (5 * w * L^4) / (384 * E * I)
- δ: Deflection at the center
- w: Uniform distributed load
- L: Length of the beam
- E: Modulus of elasticity of the beam material
- I: Moment of inertia of the beam's cross-sectional shape

3. Fixed Ends Condition:
- When both ends of the beam are fixed, it means that the beam is rigidly connected and cannot rotate at these points.
- This condition significantly increases the stiffness of the beam and reduces its deflection.

4. Comparing Deflections:
- The deflection of the simply supported beam under the distributed load is given as 100 mm at the center.
- When both ends of the beam are fixed, the deflection at the center will be reduced compared to the simply supported condition.
- Therefore, the deflection at the center when both ends are fixed will be less than 100 mm.

5. Determining the Correct Answer:
- Among the given options, the correct answer is option 'D' (2 mm), as it is the only option indicating a lower deflection compared to the simply supported condition.

In summary, when both ends of a simply supported RCC beam carrying a uniform distributed load are fixed, the deflection at the center will be reduced compared to the simply supported condition. The correct answer is option 'D' (2 mm), as it represents a lower deflection value.