Step 1: Concept Analysis:
Identify each person's colour from the puzzle solution and check which colour name lacks the letter `E'.
Step 2: Solution Approach:
Colours: Retired=Orange, Lawyer=Blue, Plumber=White, Teacher=Green, Electrician=Red, G=Silver. B is the Doctor. Doctor does not like White. G likes Silver (has E). Remaining colour for B (Doctor): Blue is Lawyer's, White is Plumber's, Green is Teacher's, Red is Electrician's, Silver is G's, Orange is Retired's. The only unassigned colour for B would be Black (no letter E). Black $\to$ no `E' $\checkmark$. Hence B's colour is Black, which does not contain the letter E.
Step 3: Conclusion:
B's favourite colour (Black) does not have the letter E in it.