The effectiveness of a fin refers to its ability to transfer heat from a solid surface to a surrounding fluid. Fins are commonly used in many engineering applications to enhance heat transfer in systems such as heat exchangers, air conditioning units, and electronic cooling devices. The effectiveness of a fin is influenced by various factors, including the type of heat transfer mechanism present in the environment.

In this case, the question asks about the effectiveness of a fin in different heat transfer environments: free convection, forced convection, radiation, and convection combined with radiation. Let's analyze each option to understand which one would result in the maximum effectiveness of a fin.

1. Free convection:
- Free convection occurs when heat is transferred through a fluid medium (such as air or liquid) due to natural buoyancy forces.
- In free convection, the fluid motion is driven solely by density differences caused by temperature variations.
- The effectiveness of a fin in free convection is generally higher compared to other mechanisms because the natural fluid motion enhances heat transfer.
- This is because the convective fluid motion helps to remove the boundary layer of stagnant fluid near the fin surface, allowing fresh fluid to come in contact with the fin and carry away heat more efficiently.

2. Forced convection:
- Forced convection occurs when heat is transferred through a fluid medium by external means such as a fan or pump.
- In forced convection, the fluid motion is artificially induced, leading to higher heat transfer rates compared to free convection.
- However, the effectiveness of a fin in forced convection is generally lower than in free convection.
- This is because the forced fluid motion tends to create a thinner boundary layer near the fin surface, reducing the contact area between the fin and the fluid and limiting heat transfer.

3. Radiation:
- Radiation is a heat transfer mechanism that does not require a medium for heat transfer.
- Instead, heat is transferred through electromagnetic waves (i.e., thermal radiation) from a hotter surface to a cooler surface.
- The effectiveness of a fin in radiation depends on the emissivity of the fin material and its surroundings.
- Generally, radiation heat transfer is less effective than convection heat transfer, especially when the temperature difference between the fin and the fluid is large.

4. Convection and radiation:
- When both convection and radiation are present, the effectiveness of a fin depends on the relative contributions of each mechanism.
- In most cases, convection dominates over radiation, especially at lower temperature differences.
- Therefore, the effectiveness of a fin in this case would be lower compared to free convection alone.

In conclusion, the correct answer to the question is option 'A' - the effectiveness of a fin will be maximum in an environment with free convection. This is because free convection enhances heat transfer by promoting fluid motion and reducing the boundary layer thickness, resulting in improved contact between the fin and the fluid.