Marta here, from Trello Support.
Right now, it's possible to delete cards by accessing the archive, which you can find on your board's menu, and deleting them from there.
Right now, it's not possible to bulk delete archived cards, so you'll need to do that one by one.
Here's an article from our help docs that gives you more details about it: http://help.trello.com/article/795-archiving-and-deleting-cards
I hope that helps!
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