Step 1: Recall how Werner's theory is actually tested experimentally.
In Werner's original experiments, a complex is dissolved and treated with $AgNO_3$. Only the chloride ions that are free (ionizable, outside the coordination sphere) precipitate immediately as $AgCl$; chloride ions bound directly to the metal inside the coordination sphere do not precipitate this way.
Step 2: Apply this to $CrCl_3 \cdot 6H_2O$.
Experimentally, all three chloride ions in this complex behave as free, ionizable ions, meaning all three would precipitate with $AgNO_3$. Ionizable chlorides balance the metal's charge, so this ionizable count equals the primary valency.
\[ x = \text{primary valency} = 3 \]
Step 3: Identify what actually sits inside the coordination sphere.
Since none of the water molecules are removed by this test, all six water molecules must be the ones directly bonded to chromium, occupying the coordination sphere.
Step 4: Secondary valency counts these directly bonded groups.
\[ y = \text{secondary valency} = 6 \]
Step 5: Coordination number is just another name for the same count.
The coordination number is the total number of donor atoms directly attached to the central metal, which is again the six water molecules.
\[ z = \text{coordination number} = 6 \]
Step 6: Final answer.
\[ x = 3,\quad y = 6,\quad z = 6 \]
\[ \boxed{(3,\ 6,\ 6)} \]