Will you fold?

Discrete Mathematics Level 5

In most five-card poker games, a hand is an unordered set of five cards. The following is the ranking of poker hands. In case an hand belongs to both possible classification of hands, only the higher rank is considered.

\(1\). Royal Flush: The set of best cards from 10 to A all in the same suit.

A♦ K♦ Q♦ J♦ 10♦

\(2.\) Straight Flush: Five Cards of Consecutive ranks of the same suit.

Q♣ J♣ 10♣ 9♣ 8♣

\(3.\) Four of a Kind: Four Cards of the same rank.

9♣ 9♠ 9♦ 9♥ J♥

\(4.\) Full House: Two Cards of the same rank and three other cards of a similar rank.

6♠ 6♥ 6♦ A♠ A♣

\(5.\) Flush: Five Cards all in the same suit

K♥ Q♥ 9♥ 5♥ 4♥

\(6.\) Straight: Five Cards of consecutive ranks

10♣ 9♦ 8♥ 7♣ 6♠.

\(7.\) Three of a kind: Three cards of the same rank

2♦ 2♠ 2♣ K♠ 6♥

\(8.\) Two Pairs: Two pairs of cards each consisting of a pair of matching ranked cards.

10♠ 10♣ 4♠ 4♥ 8♥

\(9.\) One Pair: A pair of cards of the same rank

4♥ 4♠ K♠ 10♦ 5♠

If a hand does not match any of the above, it can be called a high-card or a no-pair.

How many such high-card hands exist provided you're choosing from a standard deck?


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