you are investigating into something and researching it
What’s the story? Ask the question with the story attached to it or writing with the question
In this case the right option is the first one, <em>Neither of the children had the inclination to clean their rooms</em>. The word <em>inclination</em> is derived as stated from the Latin <em>clinare </em>and it is formed by putting together in+clinare, "lean toward [in/into]". The opposite would be "decline", formed by joining de+clinare.