The correct order of ionic radii of $Y^{3+}$, $La^{3+}$, $Eu^{3+}$, and $Lu^{3+}$ is: (Atomic nos. Y = 39, La = 57, Eu = 63, Lu = 71)

Due to the lanthanoid contraction, there is a steady decrease in the ionic radii of $M^{3+}$ ions as the atomic number increases from Lanthanum to Lutetium. Therefore, the order among the given lanthanoids is $La^{3+} > Eu^{3+} > Lu^{3+}$. Yttrium ($Y^{3+}$) belongs to the 4d transition series (Period 5) and is placed just above Lanthanum in Group 3. Its ionic radius is significantly smaller than that of $La^{3+}$ and is comparable to the heavier lanthanoids (approx. 90 pm). Although strictly speaking $Lu^{3+}$ (86 pm) is slightly smaller than $Y^{3+}$, the most appropriate sequence among the given options that correctly places $La^{3+}$ as the largest and maintains the relative order for the lanthanoids is $Y^{3+} < Lu^{3+} < Eu^{3+} < La^{3+}$.