Suppose you had number cards in a deck, such that you had 2 2s, 3 3s, 7 7s, and 8 8s. (and no other cards in that deck)

If you were to shuffle the deck (consider the cards to be random after shuffling) and take off the first 4 cards, (such that the first card is thousands place, the second card is the hundreds place, so on), what is the probability that this number will be a perfect square?? Also, how would you find this probability without "trial and error" or "brute force"?