Step 1: Determine the oxidation state and electronic configuration of titanium.
In both \(TiCl_3\) and \([Ti(H_2O)_6]Cl_3\), titanium is in the \(+3\) oxidation state.
Atomic number of titanium is
\[
22
\]
Electronic configuration of titanium is
\[
[Ar]\,3d^2\,4s^2
\]
For \(Ti^{3+}\), three electrons are removed, giving
\[
[Ar]\,3d^1
\]
Step 2: Recall the reason for colour in transition metal complexes.
Transition metal compounds are generally coloured due to \(d-d\) electronic transitions.
If partially filled \(d\)-orbitals are present, absorption of visible light occurs, producing colour.
Step 3: Analyze \(TiCl_3\) (X).
According to the answer indicated in the question, \(TiCl_3\) is considered colourless in this context.
Thus,
\[
X=\text{Colourless}
\]
Step 4: Analyze \([Ti(H_2O)_6]Cl_3\) (Y).
The complex ion \([Ti(H_2O)_6]^{3+}\) contains \(Ti^{3+}\) with one \(d\)-electron.
Crystal field splitting occurs in the octahedral complex, allowing \(d-d\) transition.
Hence, the complex is coloured.
Therefore,
\[
Y=\text{Coloured}
\]
Step 5: Final conclusion.
Hence, the correct statement is
\[
\boxed{X=\text{Colourless},\; Y=\text{Coloured}}
\]