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Given that each pair is technically the same (one 10 of hearts is not distinguishable from the other 10 of hearts), how would one calculate the number of possible hands?

You can't just do (40 choose 10) given my last statement.

I know that if I were just doing the number of possible orderings of the deck, I could do (40!)/(2!^20), but I don't think I can apply the same method to when I'm choosing a hand of 10.