Each of the castles had ITS own army of knights. The pronouns each and every always take either a singular pronoun or a singular verb. So, you can dismiss the option "their" immediately because it is a plural pronoun. So let's take a look at the other two, her and his. Castles do not really have a gender in English, and his is used for masculine gender, whereas her for feminine, so neither of those two options is correct. Thus, you have to use ITS.
