This week's book giveaway is in the Java in General forum. We're giving away four copies of Beginning Java 17 Fundamentals: Object-Oriented Programming in Java 17 and have ishori Sharan & Adam L Davis on-line! See this thread for details.
I believe what you really want is "onchange", not "onclick". If you do brute-force AJAX, your AJAX method will have to manually extract the new value from the listbox and send it to the server, and the server will need an appropriate URL handler - you can't just expect a JSF action method to be fired.
RichFaces, IceFaces, and several other extension tagsets have built-in AJAX support that can make this job much easier, including providing the ability for fire a backing bean action method or event.
Bjoke: A "Bully Joke". A Statement or action made with malicious intent - unless challenged. At which point it magically transforms into "I was just funnin'" or "What's the matter, can't take a joke?"