Deck of Cards Combinatorics

You have a deck of cards in front of you. You shuffle them completely randomly, then turn them face up. You will take two cards off in each go, and sort these two cards into one of the following piles. You decide to divide the cards into 3 piles: 1 pile will be for two cards which are both red, one pile for a pair of cards which are both black, and one pile for mismatched pairs (i.e. two cards with one red and one black).

Now, having sorted in this way through the whole deck, you notice something. What did you notice (this statement would be true for all card decks and all ways of shuffling)?

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