1. Understand the question

We need to identify the statement about lanthanoids that is not correct.

2. Check each option

Option A: All the members exhibit $+3$ oxidation state

Lanthanoids most commonly show the $+3$ oxidation state.

This is a characteristic oxidation state of the lanthanoids, so this statement is taken as correct.

Option B: Because of similar properties the separation of lanthanoids is not easy.

Lanthanoids have very similar ionic sizes and usually show the same oxidation state $(+3)$, so their chemical properties are very similar.

Hence, their separation is indeed difficult. So this statement is correct.

Option C: Availability of $4f$ electrons results in the formation of compounds in $+4$ state for all the members of the series.

This is incorrect.

Only some lanthanoids show the $+4$ oxidation state, such as:

• $Ce^{4+}$
• $Pr^{4+}$ (rare)
• $Tb^{4+}$

It is not shown by all lanthanoids.

So this statement is not correct.

Option D: There is a gradual decrease in the radii of the members with increasing atomic number in the series.

This is the phenomenon of lanthanoid contraction.

As atomic number increases, the size decreases gradually due to poor shielding by $4f$ electrons.

So this statement is correct.

3. Conclusion

The incorrect statement is:

$\boxed{C}$

4. Comparison with stored correct answer

Stored correct answer = $C$

My derived answer also = $C$

So they agree.