Multiple choice

A poker hand consists of $5$ cards drawn at random from a well-shuffled pack of $52$ cards. Then the probability that a poker hand consists of a pair and a triple of equal face values (for example, $2$ sevens and $3$ kings or $2$ aces and $3$ queens, etc.) is

  1. $\dfrac{6}{4165}$
  2. $\dfrac{23}{4165}$
  3. $\dfrac{1797}{4165}$
  4. $\dfrac{1}{4165}$
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A Correct answer
Explanation

Total ways to choose 5 cards = 52C5 = 2598960. Ways to get a full house (3 of one kind, 2 of another): Choose the rank for the triple (13C1), choose 3 suits (4C3), choose the rank for the pair (12C1), choose 2 suits (4C2). Total = 13 * 4 * 12 * 6 = 3744. Probability = 3744 / 2598960 = 6 / 4165.

AI explanation

The total number of 5-card poker hands is (52 choose 5), which equals 2,598,960. To get a pair and a triple of equal face values, choose 1 face value for the pair out of 13, then 2 cards from 4 for the pair, and 1 face value for the triple out of the remaining 12, then 3 cards from 4 for the triple. The number of successful outcomes is (13 choose 1) times (4 choose 2) times (12 choose 1) times (4 choose 3), which equals 3744. The probability is 3744 divided by 2598960, which simplifies to 6/4165.

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