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Yes you can pay by cash or part cash part card. On embarkation each passenger gets a card like a credit card which you use for all purchases on board, and then settle your bill at the end of your holiday. You can usually check the bill on your television whenever you like.
Just further to the other answer, if you elect to pay by credit card, (which can be done online well prior to sailing) you don't need to do anything to 'settle up' at the end. It will automatically appear on your cc statement,& you will usually get a printout under your door on the last night. You must also decide in what currency to pay, sterling or dollars, both of which incur some converting costs/fees.