Consider the following reaction and statements:
`[Co(NH_(3))_(4)Br_2]^(+)+Br^(-) to [Co(NH_(3))_(3)Br_3]+NH_(3)`
(I) Two isomers are produced if the reactant complex ion is a cis-isomer.
(II) Two isomers are produced if the reactant complex ion is a tran-isomer.
(III) Only one isomers is produced if the reactant complex ion is a tran-isomer.
(IV) Only one isomers is produced if the reactant complex ion is a cis-isomer.
The correct statements are :

Options

A. (III) and (IV)
B. (II) and (IV)
C. (I) and (III)
D. (I) and (IV)

Answer

To solve the problem, we need to analyze the given coordination compound reaction and the statements regarding the isomers produced. The reaction is as follows:

\[ [Co(NH_3)_4Br_2]^+ + Br^- \rightarrow [Co(NH_3)_3Br_3] + NH_3 \]

### Step 1: Identify the reactant complex ion
The reactant complex ion is \([Co(NH_3)_4Br_2]^+\). This complex can exist in two forms: cis and trans.

### Step 2: Analyze the cis isomer
1. **Cis Isomer**: In the cis configuration, the two bromide ligands are adjacent to each other.
2. When one \(NH_3\) ligand is replaced by a \(Br^-\) ion, two types of isomers can be formed:
- **Facial Isomer**: All three \(Br\) ligands are adjacent to each other.
- **Meridional Isomer**: The \(Br\) ligands are arranged such that two are opposite each other, and one is adjacent.

Thus, **two isomers are produced from the cis isomer**.

### Step 3: Analyze the trans isomer
1. **Trans Isomer**: In the trans configuration, the two bromide ligands are opposite each other.
2. When one \(NH_3\) ligand is replaced by a \(Br^-\) ion, only one type of isomer can be formed:
- **Meridional Isomer**: Since the \(Br\) ligands are already opposite each other, replacing any \(NH_3\) will lead to a configuration that results in only the meridional isomer.

Thus, **only one isomer is produced from the trans isomer**.

### Step 4: Evaluate the statements
- **Statement I**: True - Two isomers are produced if the reactant complex ion is a cis-isomer.
- **Statement II**: False - Two isomers are not produced if the reactant complex ion is a trans-isomer (only one isomer is produced).
- **Statement III**: True - Only one isomer is produced if the reactant complex ion is a trans-isomer.
- **Statement IV**: False - Only one isomer is produced if the reactant complex ion is a cis-isomer (not true).

### Conclusion
The correct statements are I and III.

### Final Answer
The correct statements are: **I and III**.