Consider that the coordinating atoms of the ligands in cis-[Co(NH₃)₄Cl₂]Cl and mer-[Co(NH₃)₃Cl₃] octahedral complexes are at the vertices of an octahedron. The sum of total number of the triangular faces in both the complexes having one N atom and two Cl atoms at their corners is _______.

The correct answer is 6.

Let us analyze the structure of octahedral complexes and determine the number of triangular faces having one nitrogen atom (from NH₃) and two chlorine atoms (from Cl) at their vertices for both given complexes.

An octahedron has 6 vertices and 8 triangular faces. In an octahedral complex, the 6 coordinating ligand atoms occupy these 6 vertices.

Let the 6 vertices of the octahedron be represented in Cartesian coordinates as:
+x, -x, +y, -y, +z, -z.
The 8 triangular faces of an octahedron are formed by combinations of three mutually adjacent (cis) vertices:
1. (+x, +y, +z)
2. (+x, +y, -z)
3. (+x, -y, +z)
4. (+x, -y, -z)
5. (-x, +y, +z)
6. (-x, +y, -z)
7. (-x, -y, +z)
8. (-x, -y, -z)

1. Complex 1: cis-[Co(NH₃)₄Cl₂]Cl

In the octahedral coordination sphere [Co(NH₃)₄Cl₂]⁺, there are 4 N atoms (from four NH₃ ligands) and 2 Cl atoms (from two Cl^- ligands).

Since it is the cis isomer, the two Cl atoms are adjacent to each other (at a 90° angle). Let us place the two Cl atoms at vertices:
Cl₁ at +x
Cl₂ at +y
The remaining 4 positions (-x, -y, +z, -z) are occupied by 4 N atoms (N₁, N₂, N₃, N₄).

We need to find the number of triangular faces that contain exactly one N atom and two Cl atoms.
A face with two Cl atoms must include both the (+x) and (+y) vertices. Looking at the 8 faces:
- Face (+x, +y, +z): Vertices are Cl₁, Cl₂, and N (at +z). This face has 2 Cl atoms and 1 N atom.
- Face (+x, +y, -z): Vertices are Cl₁, Cl₂, and N (at -z). This face also has 2 Cl atoms and 1 N atom.
None of the other 6 faces contain both (+x) and (+y).

Therefore, for cis-[Co(NH₃)₄Cl₂]Cl, the number of such triangular faces = 2.

2. Complex 2: mer-[Co(NH₃)₃Cl₃]

In mer-[Co(NH₃)₃Cl₃], the three Cl atoms lie on a meridian plane passing through the central Cobalt atom. That means two Cl atoms are trans to each other (180° apart) and the third Cl atom is cis to both of them.

Let us assign the positions of the 3 Cl atoms as:
Cl₁ at +x
Cl₂ at -x (trans to Cl₁)
Cl₃ at +y (cis to both Cl₁ and Cl₂)
The 3 N atoms are placed at the remaining positions: -y, +z, -z.

We need to find the number of triangular faces containing two Cl atoms and one N atom:
Pair of adjacent Cl atoms can be:
- (Cl₁, Cl₃) at (+x, +y)
- (Cl₂, Cl₃) at (-x, +y)
(Note: Cl₁ and Cl₂ are trans to each other, so they do not share any triangular face).

Let us examine the faces containing these adjacent Cl pairs:
- Faces with (+x, +y):
1. (+x, +y, +z) → Cl₁, Cl₃, N (at +z) [Contains 2 Cl, 1 N]
2. (+x, +y, -z) → Cl₁, Cl₃, N (at -z) [Contains 2 Cl, 1 N]
- Faces with (-x, +y):
3. (-x, +y, +z) → Cl₂, Cl₃, N (at +z) [Contains 2 Cl, 1 N]
4. (-x, +y, -z) → Cl₂, Cl₃, N (at -z) [Contains 2 Cl, 1 N]

Thus, for mer-[Co(NH₃)₃Cl₃], the number of such triangular faces = 4.

Conclusion:

Sum of total number of triangular faces having one N atom and two Cl atoms in both complexes:
$\text{Sum}=2+4=6$