Write the expected electronic filling for Gadolinium, $Z=64$

Gadolinium comes after Xenon, so its inner core is: $[Xe]$

After $[Xe]$, electrons enter the $6s$, $4f$, and sometimes $5d$ subshells.

Recall the lanthanide pattern

In the lanthanide series, the general filling is close to: $[Xe]4f^n 5d^{0\text{ or }1} 6s^2$

Gadolinium is a special case because it achieves extra stability with a half-filled $4f$ subshell.

Find the stable configuration of Gd

For Gd ($Z=64$), the configuration is: $[Xe]4f^7 5d^1 6s^2$

Check total electrons beyond xenon: $7+1+2=10$ Since Xenon has $54$ electrons, $54+10=64$ which matches Gd.

Compare with the options

• A: $4f^8 5d^0 6s^2$
  Not the accepted ground-state configuration for Gd.

• B: $4f^4 5d^4 6s^2$
  Incorrect.

• C: $4f^7 5d^1 6s^2$
  Correct.

• D: $4f^3 5d^5 6s^2$
  Incorrect.

Final answer

The outer electronic configuration of Gd is: $\boxed{4f^7 5d^1 6s^2}$

So, the correct option is C.