A list of poker hands ranked in order from the highest to the lowest: Different hands for a three of a kind?

Question by : A list of poker hands ranked in order from the highest to the lowest: Different hands for a three of a kind?
If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, how many different hands consist of the following: Three of a kind?
How many different hands??

Best answer:

Answer by WOW ARE WE STUPID OR WHAT
Three of a kind, also called trips or a set, is a poker hand such as 2♦ 2♠ 2♣ K♠ 6♥ that contains three cards of the same rank, plus two cards which are not of this rank nor the same as each other. In Texas hold ’em and other flop games, three of a kind is called a “set” usually when it is composed of a pocket pair and one card of matching rank on the board;[3] It is called “trips” usually when it is made by one card that player has in the hole with two matching cards on the board.[4]

Higher-valued three of a kind defeat lower-valued three of a kind, so Q♠ Q♥ Q♦ 7♠ 4♣ defeats J♠ J♣ J♦ A♦ K♣. If two hands contain three of a kind of the same value, which is possible in games with wild cards or community cards, the kickers are compared to break the tie, so 4♦ 4♣ 4♠ 9♦ 2♣ defeats 4♦ 4♣ 4♠ 8♣ 7♦.

There are 54,912 possible three of a kind hands that are not also full houses or four of a kind; the probability of being dealt one in a five-card deal is \frac {C_{13}^1 C_4^3 \cdot C_{12}^2 C_4^1 C_4^1} {C_{52}^5} = \frac {13 \cdot 4 \cdot 66 \cdot 4 \cdot 4} {2{,}598{,}960} = \frac {54{,}912} {2{,}598{,}960} \approx 2.11%

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