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Since it's video Poker, we can assume that the only cards drawn are the ones you started with. There are exactly 47 cards left in the deck. There are 6 series of draws you can get that will give you ace, queen, and 10 (3!, or 3x2x1).

The number of sequences of 3 cards in a 47 card deck is 47!/44!, or 47x46x45, and since 6 of the sequences leave you with the hand you want, you have exactly 6/(47x46x45) probability to get one of them.

This works out to 1 in 16,215.

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