The number of optical isomers possible for [Cr(C$$_2$$O$$_4$$)$$_3$$]$$^{3-}$$ is:

We begin by identifying all the basic facts about the ion $$\left[\text{Cr}\left(\text{C}_2\text{O}_4\right)_3\right]^{3-}.$$ The central metal is chromium in the +3 oxidation state, denoted $$\text{Cr}^{3+}.$$ A chromium(III) ion has the electronic configuration $$[ \text{Ar} ]\,3d^3.$$ With three $$d$$ electrons, it normally forms an octahedral complex because six positions around the metal are available for coordination.

Each oxalate ion, written $$\text{C}_2\text{O}_4^{2-},$$ is a bidentate ligand. “Bidentate” means it donates two lone-pair electrons, one from each of two oxygen atoms, and therefore occupies two adjacent coordination sites on the metal. Since one oxalate occupies two sites, three oxalate ions exactly satisfy the six available positions of the octahedron:

$$3 \times 2 = 6 \; \text{(coordination sites)}.$$

Whenever a single ligand uses two donor atoms to bind to the same metal, it forms a chelate ring. Here, each oxalate forms a five-membered ring with chromium. Because all three ligands are identical and wrap around the metal, the resulting shape is rigid and can be chiral.

For an octahedral complex of general type $$[\text{M}(AA)_3],$$ where $$AA$$ is a symmetrical bidentate ligand such as oxalate, a well-known stereochemical rule applies:

“A complex of the form $$[\text{M}(AA)_3]$$ gives exactly two non-superimposable mirror images, called the $$\Delta$$ (right-handed) and $$\Lambda$$ (left-handed) forms.”

These two forms are enantiomers. They are mirror images that cannot be rotated to overlap; nevertheless, they contain no plane or centre of symmetry, so they are optically active. There are no further distinct stereoisomers because:

• All three ligands are the same, eliminating the possibility of facial/meridional or cis/trans arrangements.
• Each bidentate ligand must occupy adjacent sites, so no linkage isomerism arises.
• No internal mirror plane or inversion centre exists in either of the two arrangements, confirming chirality.

Therefore the number of possible optical isomers—i.e. the number of distinct chiral structures—is exactly the size of this enantiomeric pair:

$$\text{Number of optical isomers} = 2.$$

So, the answer is $$2$$.