1. We need to count the species that are non-pyramidal among:

   • $SO_3$
   • $NO_3^-$
   • $PCl_3$
   • $CO_3^{2-}$

2. Recall:

   • Pyramidal shape usually means trigonal pyramidal, which occurs when the central atom has $3$ bond pairs and $1$ lone pair.
   • If a species is trigonal planar, it is non-pyramidal.

3. Now analyze each species using VSEPR theory.

(A) $SO_3$

• Central atom: $S$
• Around sulfur, there are three regions of electron density due to three $S-O$ bonds.
• No lone pair on sulfur in the usual VSEPR description.
• Shape = trigonal planar.

So, $SO_3$ is non-pyramidal.

(B) $NO_3^-$

• Central atom: $N$
• Three $N-O$ bonds around nitrogen.
• No lone pair on nitrogen.
• Shape = trigonal planar.

So, $NO_3^-$ is non-pyramidal.

(C) $PCl_3$

• Central atom: $P$
• Phosphorus has $5$ valence electrons.
• It forms three $P-Cl$ bonds and retains one lone pair.
• Thus, there are $4$ electron pairs total: tetrahedral electron geometry.
• Molecular shape = trigonal pyramidal.

So, $PCl_3$ is pyramidal, hence not counted.

(D) $CO_3^{2-}$

• Central atom: $C$
• Three $C-O$ bonds around carbon.
• No lone pair on carbon.
• Shape = trigonal planar.

So, $CO_3^{2-}$ is non-pyramidal.

4. Counting non-pyramidal species:

• $SO_3$ : yes
• $NO_3^-$ : yes
• $PCl_3$ : no
• $CO_3^{2-}$ : yes

Total number of non-pyramidal species $= 3$.

$\boxed{3}$

5. Comparison with stored correct answer:

• Stored correct answer = $3$
• Derived answer = $3$
• They match.