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I assume we are speaking of Pres. Obama and his family, but the response would still apply if we were speaking of the President of another country. As a native-born American, I believe I can speak with some authority on this matter. When we speak of the First Family, we mean the US President, his wife and his children. However, this query deals with the President and his family separately, so we are talking about just his wife and children, i.e., keluarganya. On Google, there are 35,200 hits on "Presiden Obama dan keluarga", and 18,600 hits on "Presiden Obama dan keluarganya". Either one would suffice, but grammatically, "... keluarganya" would be lebih baku.

Disebut first family karena mereka ini keluarganya the first person.