<p>In a poker game what is the probability of getting a "pair" in five cards? [A pair consists of 2 cards of the same kind (e.g., 2 Kings) and 3 cards that are different from the kind of the pair (e.g., different from Kings) and that are all different from each other.]</p>
Step-by-Step Solution
Key Concept: A pair requires exactly 2 cards of one rank and 3 cards of different ranks (all distinct from each other and the pair). Use combinations to count favorable outcomes: choose rank for pair, choose 2 from 4 cards of that rank, choose 3 different ranks from remaining 12, then choose 1 card from each of those 3 ranks.
<p><strong>Step 1:</strong> Total ways to choose 5 cards from 52: $\binom{52}{5} = 2,598,960$</p><p><strong>Step 2:</strong> Count favorable outcomes for exactly one pair:</p><ul><li>Choose rank for the pair: 13 ways</li><li>Choose 2 cards from 4 of that rank: $\binom{4}{2} = 6$ ways</li><li>Choose 3 different ranks from remaining 12 ranks: $\binom{12}{3} = 220$ ways</li><li>Choose 1 card from each of these 3 ranks: $4 \times 4 \times 4 = 64$ ways</li></ul><p><strong>Step 3:</strong> Favorable outcomes = $13 \times 6 \times 220 \times 64 = 1,098,240$</p><p><strong>Step 4:</strong> Probability = $\frac{1,098,240}{2,598,960} = \frac{352}{833} \approx 0.423$ or about 42.3%</p><p>∴ Answer: B</p>
Correct Answer: B