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Depends on how young. Whilst many cater for children, some of the children's facilities (like creches etc) have minimum ages. I've heard that RCI have lots of activities for children. MSC offer free child places, but Brits arte often in the moinority. Other cruise lines do child discounts. Either have a look at a few of the cruise lines websites to see what facilities they offer and how they charge for childreen or call a cruise specialist, such as the tel number on this sit.
In my experience Royal Caribbean have the most activities for children...especialy the larger Voyager class or Freedom class ships.