Well, the basic idea is to create or use an encryption scheme where, without knowing exactly what each encrypted thing is, you can count and identify unique records.
Take your playing cards. First off, without looking at any card's face, you can easily count them and verify that they at least came from the same style of deck. This is because each individual card is countable and identifiable as belonging to a deck of a particular style. In terms of data, you would get the data as elements of a list, delimited or otherwise divisible into individual elements, and those elements would have some sort of header you can use to identify the items as belonging to some unique set. However, you could be holding a Pinochle deck with 4 Jokers added, and not a Poker deck, and not know the difference. To be able to tell the difference, you have to know something about the values, without knowing the values.
There are two basic types of encryption that would allow you to accomplish this. One is a one-way hash; the values are translated, theoretically irreversibly, into a form that can be compared for equality but nothing else. If each hash is unique, you can be confident that the makeup of the deck is close enough to a Poker deck that it can be used as one. The other is a "block cipher"; the same key, plugged into the same algorithm, is used to transform each iterative piece of the plaintext into the ciphertext. Theoretically, without knowing the key, you can't discern the plaintext, but assuming that each card's value is encrypted in this way, each ciphertext should be unique; duplicate ciphertexts indicate duplicate plaintexts and thus duplicate cards. The actual ciphering scheme is irrelevant - you could use a Caesar cipher or AES - what's important is that each card's value is independently encrypted in exactly the same way, using no other information than one card's value, the key and the encryption scheme.
EDIT: The last thing you would have to prove is that, with all cards being unique, each one is a card you would expect to find in a poker deck (that is, there is no Zero of Spades). This is the tricky part, and IMHO you can't do it in a Zero-Knowledge way. you would have to know something about how the ciphertexts were encrypted in order to identify ciphertexts that were valid and ones that weren't.