In the full decks, John has a total of 52 cards in a full deck of cards * 3 full decks of cards = <<52*3=156>>156 cards.
A half-full deck of cards contains 52 cards in a full deck of cards / 2 = <<52/2=26>>26 cards.
So from the half-full decks, John has a total of 26 cards * 3 half-full decks = <<26*3=78>>78 cards.
This is a total of 156 cards from full decks + 78 cards from half-full decks = <<156+78=234>>234 cards.
After throwing some in the trash, he now holds a total of 234 cards - 34 trashed cards = <<234-34=200>>200 cards.
#### 200