A poker hand is defined as five cards randomly drawn from a standard 52-card deck. Note that each 52-card deck has 13 ranks (two, three,…,ten, jack, queen, king, ace) and each rank is represented in 4 suits (hearts, diamonds, clubs, and spades).

The probabilities of the various poker hands are shown below. Note there may be multiple, correct solution methods (not all listed). If you use a different method, compare the number of possibilities in your numerator with the number of possibilities listed in the numerator of these solutions. If the numbers are the same, then the solution methods are equivalent.

Royal Flush (ten, jack, queen, king, and ace all of one suit)

Straight Flush (five cards in order and all of the same suit—note that an ace can be used as a high card or a low card, and a royal flush doesn’t count as a “straight flush”)

Four of a Kind (four cards of the same rank)

Full House (three cards of one rank and two cards of another rank)

Flush (five cards of the same suit—note that a royal flush and a straight flush don’t count as a “flush”)

Straight (five cards in order—note that an ace can be used as a high or low card, and a royal flush and a straight flush don’t count as a “straight”)

Three of a Kind (exactly three cards of the same rank—note that a full house doesn’t count as a “three of a kind”)

Two Pair (exactly two cards of one rank and exactly two cards of another rank)

One Pair (exactly two cards of one rank)