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26 of the 52 cards are black.

For each of those cards, the second card in your hand may not be the same card, but it may be any of the 25 remaining ones.

The third card, likewise, can not be the first two, but can be any of the 24 remaining.

For a four card hand that's 26 * 25 * 24 * 23 = 358800 possible hands

However, that number includes sets that are effectively duplicates (1,2,3,4 and 4,3,2,1 are both accounted for in that number, but for our purposes those are the same hand).

We can arrange each set of cards 4 * 3 * 2 * 1 = 24 ways. So to remove those possibilities we take 358800/24 = 14950 possible hands of all black cards.

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Michael Scalise

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