The common oxidation state of the elements in the lanthanoids series is 3.

Explanation:
The lanthanoids, also known as the lanthanides, are a series of elements in the periodic table that include atomic numbers 57 to 71. These elements are characterized by the filling of the 4f electron shell.

Electron Configuration:
The electron configuration of the lanthanoids involves the filling of the 4f orbital. The general electron configuration for the lanthanoids is [Xe]4f^n6s^2, where n represents the number of electrons in the 4f orbital.

Stability of Oxidation States:
The stability of oxidation states depends on the electronic configuration of the elements. In the case of the lanthanoids, the 4f electrons are shielded by the 5s and 5p electrons. As a result, the 4f electrons are less available for bonding, and the elements tend to exhibit a common oxidation state of +3.

Reasoning:
The common oxidation state of +3 for the lanthanoids can be attributed to the stability of the 4f electron configuration with a completely filled or half-filled 4f subshell. This stability arises due to the presence of Hund's rule, which states that electrons tend to occupy separate orbitals of the same energy before pairing up.

Exceptions:
There are a few exceptions to the common +3 oxidation state in some lanthanoids. For example, cerium (Ce) can exhibit an oxidation state of +4 in certain compounds, while europium (Eu) and ytterbium (Yb) can exhibit oxidation states of +2 and +2/+3, respectively.

Applications:
The lanthanoids find applications in various fields, including catalysts, magnets, lasers, and lighting. Their ability to exhibit multiple oxidation states makes them versatile for different chemical reactions and properties.

In conclusion, the common oxidation state of the elements in the lanthanoids series is +3, which is due to the stability of the 4f electron configuration. However, there are exceptions in some lanthanoids where other oxidation states are possible.