The adjective each is always followed by a singular noun: each person, each book. When the adjective follows a plural subject, the verb agrees with the subject: the children each have many books. When the pronoun each comes immediately before the verb, it takes a singular verb: Each lives in a different neighborhood. When the pronoun is followed by a phrase containing a plural noun, the verb can be plural: Each of them have lovely homes.
Anyone, anybody, everyone, everybody, no one, someone, and somebody follow the same general patterns of pronoun agreement as each and none. However, these are not hard and fast rules anymore but guidelines for formal writing. For dialogue or other forms of casual writing, it is not necessary to concern yourself with these rules.
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