More number of oxidation states is exhibited by the actinoids than by the lanthanoids. The main reason for this is

Lower energy difference between 5f and 6d orbitals than between 4f and 5d orbitals

The correct answer is option (4) Lower energy difference between 5f and 6d orbitals than between 4f and 5d orbitals.

The greater number of oxidation states exhibited by actinoids compared to lanthanoids is primarily attributed to the electronic configurations of their valence electrons. In actinoids, the 5f and 6d orbitals are very close in energy, resulting in significant overlap and mixing of these orbitals. This overlap allows for greater flexibility in the oxidation states that actinoids can adopt. On the other hand, in lanthanoids, while the 4f and 5d orbitals are also close in energy, the energy difference between them is greater compared to that between the 5f and 6d orbitals in actinoids. This larger energy difference limits the number of oxidation states available to lanthanoids. Therefore, the lower energy difference between the 5f and 6d orbitals in actinoids allows for a wider range of oxidation states, leading to the observation of more oxidation states in actinoids compared to lanthanoids.

The other options are incorrect.

Option (1): The greater metallic character of the lanthanoids is not the reason why they exhibit fewer oxidation states. In fact, the actinoids are also metals, and they have similar metallic properties to the lanthanoids.

Option (2): The more active nature of the actinoids is not the reason why they exhibit more oxidation states. In fact, the lanthanoids are also active metals, and they can react with a variety of elements.

Option (3): The more energy difference between the 5f and 6d orbitals than between 4f and 5d orbitals would mean that it is harder for electrons to be promoted from the 5f orbitals to the 6d orbitals in the actinoids than in the lanthanoids. This would result in the actinoids exhibiting fewer oxidation states, not more.

Therefore, the answer is (4). The lower energy difference between the 5f and 6d orbitals than between the 4f and 5d orbitals is the reason why the actinoids exhibit more oxidation states than lanthanoids.