Poker hands
Probability
Level
2
From a deck of 52 cards, how many 5 card hands can be formed with at least 3 face cards? Find the Probability
4.6%
8.9%
7.3 %.
7.6%
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Total 5 card hands = 52 × 51 × 50 × 49 × 48 / 5!
There are 12 face cards in the deck (J Q K of each suit), and 40 other cards.
We can build a hand with 3 face cards by choosing the face cards first, then the others. That can be done in 12 × 11 × 10 × 40 × 39 / (3! × 2!) ways. Similarly, for hands with 4 face cards, there are 12 × 11 × 10 × 9 × 40 / 4! ways. And for 5 face cards, 12 × 11 × 10 × 9 × 8 / 5! ways. Adding all those up, we have 192192 ways to have 3 or more face cards. Out of the total 2598960 hands, the probability is around 7.3 %.