Total number of geometrical isomers of `[CoBrClI(CN)(H_(2)O)(NH_(3))]^(-)` complex ion, in which all halides are in cis-position.

Explanation

To determine the total number of geometrical isomers of the complex ion \([CoBrClI(CN)(H_2O)(NH_3)]^{-}\) with all halides in cis position, we can follow these steps:

### Step 1: Identify the coordination number and geometry
The coordination number of cobalt in this complex is 6, as it is surrounded by six ligands (3 halides and 3 other ligands). The geometry of a coordination complex with a coordination number of 6 is typically octahedral.

**Hint:** Remember that the geometry of the complex is determined by the coordination number.

### Step 2: Determine the fixed positions of the halides
Since the problem states that all halides (Br, Cl, and I) are in cis positions, we can visualize them occupying adjacent positions in the octahedral structure. For example, we can place Br and Cl in cis positions, and I will also be in a cis position relative to them.

**Hint:** In an octahedral complex, cis positions are those that are adjacent to each other.

### Step 3: Identify the other ligands
The remaining three ligands (CN, H2O, NH3) can occupy the remaining three positions in the octahedral structure. Since these ligands are not fixed in position, they can be arranged in different ways.

**Hint:** The arrangement of the remaining ligands will contribute to the different geometrical isomers.

### Step 4: Count the arrangements of the remaining ligands
The three remaining ligands (CN, H2O, NH3) can be arranged in the three available positions. The number of ways to arrange 3 different ligands is given by the factorial of the number of ligands, which is \(3!\).

\[ 3! = 6 \]

**Hint:** Use the factorial function to count the arrangements of distinct items.

### Step 5: Conclusion
Since all halides are fixed in cis positions, and the three remaining ligands can be arranged in 6 different ways, the total number of geometrical isomers for the complex ion \([CoBrClI(CN)(H_2O)(NH_3)]^{-}\) is 6.

**Final Answer:** The total number of geometrical isomers is 6.