A full house is a poker hand with 3 cards of one denominatio

A full house is a poker hand with 3 cards of one denomination and 2 cards of another, for example, three kings and two jacks. Show that four of a kind beats a full house.

Solution

There are 13 ways to choose a four of a kind, and 48 ways to choose the other one card.

Hence, n(four of a kind) = 13*48 = 624

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There are 13 ways to choose the rank of your trio, and 4C3 = 4 ways to choose the trio in that rank.

There are a remaining of 12 ways to choose the rank of the next pair, with 4C2 = 6 ways to choose the pair in that rank.

Thus, n(full house) = 13*4*12*6 = 3744.

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As we saw, four of a kind is RARER than a full house, hence, it beats full house. [CONCLUSION]