Question

Among the 2,598,960 possible 5-card poker hands from a standard 52-card deck, how many contain:

a) Four of a kind
b) Two pairs
c) A flush
d) Exactly three of a kind

Answer 1

The number of possible outcomes for 5-card poker hands is 52 to the power of 5. There are 13 * 48 four of a kind hands, 13 * 13 * 44 * 4 two pair hands, 4 * 13 flush hands, and 13 * 12 * 12 * 4 hands with exactly three of a kind.

Step-by-step explanation:

The number of possible outcomes (microstates) of any repeated independent situation is equal to the number of possibilities in one iteration to the power of the repetitions. So in this case, the answer is 52 to the power of 5.

For a), to calculate the number of four of a kind hands, we need to consider the rank of the four cards (13 options) and the rank of the remaining card (48 options). So the total number of four of a kind hands is 13 * 48.

For b), to calculate the number of two pair hands, we need to consider the ranks of the two pairs (13 options each), the rank of the fifth card (44 options), and the suit of the five cards (4 options). So the total number of two pair hands is 13 * 13 * 44 * 4.

For c), to calculate the number of flush hands, we need to consider the suit of the five cards (4 options) and the rank of the five cards within the suit (13 options). So the total number of flush hands is 4 * 13.

For d), to calculate the number of hands with exactly three of a kind, we need to consider the rank of the three cards (13 options), the rank of the remaining two cards (12 options each), and the suits of the five cards (4 options). So the total number of hands with exactly three of a kind is 13 * 12 * 12 * 4.