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I'm trying to figure out the probability that the game ends in 5 turns, 6, 7, 8, ... 21, but my knowledge of factorials, which I suspect could help here, is limited (B+ in college calculus some years ago).

I know 5: Any card can be drawn first, then the probabilities are 20/24 * 15/23 * 10/22 * 5/21 = 15,000/255,024 = .0588 = 5.88%. So there is a 5.88% chance the game ends when the 5th card is revealed. Easy enough.

By following a similar process I believe there is an 11.75% of the game ending in 6 rounds, but I'm not sure that answer is correct and I'm looking for a formula that might simplify things.

Your thoughts welcome.