1. Identify the geometry and possible isomers

The complex ion is $[Co(NH_3)_4Br_2]^+$ which is an octahedral complex of type $MA_4B_2$.

Such complexes show cis and trans isomerism.

The product is $[Co(NH_3)_3Br_3]$ which is an octahedral complex of type $MA_3B_3$.

Such complexes have two geometrical isomers:

• fac
• mer

So, we need to see whether substitution of one $NH_3$ by $Br^-$ from the given reactant leads to one or two possible product geometries.

2. Case 1: Reactant is cis-$[Co(NH_3)_4Br_2]^+$

In the cis isomer, the two $Br$ ligands are adjacent. After replacement of one $NH_3$ by $Br^-$, the product has three $Br$ ligands total.

Now consider the positions available for the incoming $Br$ relative to the two already present adjacent $Br$ ligands:

• If the new $Br$ occupies the position cis to both existing $Br$ ligands, then all three $Br$ ligands lie on one face of the octahedron, giving the fac isomer.
• If the new $Br$ occupies a position such that one pair becomes trans, then the arrangement becomes mer.

Thus, from the cis reactant, two isomers of $[Co(NH_3)_3Br_3]$ are possible.

So, statement (I) is correct.

And therefore statement (IV) is false.

3. Case 2: Reactant is trans-$[Co(NH_3)_4Br_2]^+$

In the trans isomer, the two $Br$ ligands are opposite to each other. After substitution of one $NH_3$ by $Br^-$, there are three $Br$ ligands total.

Since two bromides are already trans, in the product there must remain a trans pair among the three bromides. That corresponds only to the mer arrangement.

The fac isomer cannot be formed because in fac-$MA_3B_3$, all three identical ligands are mutually cis; there is no trans pair.

Hence, from the trans reactant, only one isomer is produced.

So, statement (III) is correct.

And therefore statement (II) is false.

4. Evaluate the statements

• (I) Two isomers are produced if the reactant complex ion is a cis-isomer $\rightarrow$ True
• (II) Two isomers are produced if the reactant complex ion is a trans-isomer $\rightarrow$ False
• (III) Only one isomer is produced if the reactant complex ion is a trans-isomer $\rightarrow$ True
• (IV) Only one isomer is produced if the reactant complex ion is a cis-isomer $\rightarrow$ False

Therefore, the correct combination is: $\boxed{\text{(I) and (III)}}$

So the correct option is: $\boxed{\text{C}}$