Family trip to New York

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I want to take my family to New York next year for Christmas. I'd love for them to see the lighting of the tree in Rockefeller center, the museums, shows on Broadway, Statue of Liberty and anything else associated with New York but I wouldn't exactly want to stay in a hotel in the city. I'd rather rent a house where we can spread out at the end of the day. When we went to Orlando we found it cheaper to rent a house. Would it be the same in this situation? Is it even possible?