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I understand that the word "family" is generally singular, as the word "families" is plural. I also know that the word "family" is a collective noun. My question is: Should the sentence read, "The new family moved into his or her house?" or "The new family moved into their house?"
If it is "their" doesn't this indicate that "family" is plural?
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In British English, 'family' is often treated as plural.

My family have given up Christmas presents.
John's family are not being invited to his wedding.
jediguardianShould the sentence read, "The new family moved into his or her house?" or "The new family moved into their house?"
His and her are definitely wrong if the possessives refer to the family. Even people who use a singular verb in my examples above would use'their' in your sentence.
jediguardianIf it is "their" doesn't this indicate that "family" is plural?
No necessarily, 'Their' is being increasingly accepted as a gender-neutral singular form. I saw somebody on the bus I thought I recognised, but I couldn't remember their name.
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The sort of combination that I personally dislike is

The family has moved into their new home.
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Thank you for the clarity. I appreciate the response.
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GPYThe sort of combination that I personally dislike is The family has moved into their new home.
Do you like (or say or recommend) its new home? Emotion: surprise

CJ
CalifJimDo you like (or say or recommend) its new home?
In most circumstances I would advocate "The family have moved into their new home". "The family has moved into its new home", while perfectly correct in my opinion, tends to sound distant and formal.