Final answer:
a. philately
Philately is the nine-letter word that describes the hobby of collecting stamps; it is distinct from numismatics, archaeology, and calligraphy.
Step-by-step explanation:
The hobby of collecting stamps is known as philately. This nine-letter word is an activity that involves collecting and studying stamps, postmarks, and related postal materials. Philately is distinct from numismatics, which is the study or collection of currency, including coins, tokens, paper money, and related objects. It's also not to be confused with archaeology, which is the study of human history and prehistory through the excavation of sites and the analysis of artifacts and other physical remains, nor with calligraphy, which is the art of beautiful handwriting.