Handsomedan:
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In saying that, once cut in half, a card is 100% unusable. Also if it's expired, the card is no longer valid in a card present situation and in a card not present environment (i.e. online) the new expiry date and CVV are not going to line up, so again it's useless.
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How then does a company (Dropbox in this case), manage to charge a current card, when all they hold are the details for an expired card?