Step 1: Understand the question.
We must find the pair that does NOT have the same number of electrons. Species with equal electron counts are called isoelectronic. So we just count electrons in each species and look for the odd pair out.
Step 2: Recall the counting rule.
For an atom, electrons equal the atomic number $Z$. For a positive ion, subtract the charge; for a negative ion, add the charge. This is because removing electrons gives a positive ion and adding electrons gives a negative ion.
Step 3: Check $\mathrm{Na}^+$ and $\mathrm{Al}^{3+}$.
Sodium ion: $11-1=10$. Aluminium ion: $13-3=10$. Both have $10$, so they match.
Step 4: Check $\mathrm{O}^{2-}$ and $\mathrm{F}^{-}$.
Oxide ion: $8+2=10$. Fluoride ion: $9+1=10$. Both have $10$, so they match too.
Step 5: Check $\mathrm{Mg}^{2+}$ and $\mathrm{Ar}$.
Magnesium ion: $12-2=10$. Argon atom: $18$. These are $10$ and $18$, which are different. This is our odd pair.
Step 6: Confirm with $\mathrm{P}^{3-}$ and $\mathrm{Ar}$.
Phosphide ion: $15+3=18$. Argon: $18$. These match, so they are not the answer. Hence the pair that is not equal is $\mathrm{Mg}^{2+}$ and $\mathrm{Ar}$, option 3. \[ \boxed{\mathrm{Mg}^{2+}\ \text{and}\ \mathrm{Ar}} \]