Techniques to improve reliability - Part 11
Clue 4: Professor Plum was not in the library nor the billiard room. This clue is not relevant.
Clue 5: The person with the candlestick was in the observatory. This clue is relevant.

(2) Second, combine the relevant clues to reason out the answer to the question:
Clue 3 and Clue 5 both say that the person with the candlestick was in the observatory. Colonel Mustard was the only person in the observatory, so Colonel Mustard was the person with the candlestick.