Some might think a predicate nominative is used in this sentence. But nominatives are only used when there is a linking verb, "to be." An example of predicate nominative is: "This is he." Another example is: "It is I." With the verb "to be," the object is in the nominative case.
In the above example, the verb is "included." Therefore, the sentence does not use a predicate nominative. This simply means that the object of the verb is in the accusative case. So, the sentence should read:
"Famous guests at the banquet included Archie, Ray, and him."
This is actually a very straightforward question. To simplify things, you can write, "Famous guest at the banquet included him." "He" would not work at all, because it is used as a subject pronoun.
The correct answer to this question is 'him' because it is part of the Object of the Verb, in the accusative case, necessitating the use of an object pronoun. The word 'included' in this case is a verb that is transitive, requiring an Object. The word 'him' fills the object slot of the transitive verb 'included'.
The reason why 'him' is the only choice for this sentence is because of the nature of pronouns. Some pronouns are subject pronouns: I, we, they, he, who. These subject pronouns are used to substitute in the sentence for the Subject. Other pronouns are object pronouns: me, us, them, him, whom. These object pronouns must be used to fill object slots, as subject pronouns are incorrect when used as the object of verbs.