More than one missing card from a deck

Hey everybody! I’m new here, and I’m specifically looking for a method to identify multiples of missing cards from a non-uniform, 60 card deck of non-playing cards (there can be be between 1 and 4 copies of any card in the deck) for a trading card game. The player knows the full contents of the deck before the exercise starts.
I find the problem more intriguing if the number of missing cards can vary, but in practice, 6 cards out of the 60 are removed from the deck and placed out of play. Right now, I find myself having to go through the deck 5 or 6 times, each time counting a small set of cards and remembering what is missing in order to figure out which 6 cards have been removed, but this is too slow.
I haven’t been able to find any methods for identifying more than one missing card in a deck of playing cards, but if you know of any, feel free to share those as well!

Would this technique work for you?

  1. Memorize the deck using a memory palace.
  2. Mentally walk through your full set of 60 card images in order, making note of the ones that aren't in that memory palace. Put those images in a second, smaller palace.
  3. Read off the cards in the second palace to name the missing cards.