Question

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

• A. $\frac{6}{4165}$
• B. $\frac{23}{4165}$
• C. $\frac{1797}{4165}$
• D. $\frac{1}{4165}$

Answer 1

Answer: (A)

$\therefore $ Required probability
$=\frac{{ }^{13} C_{1} \times{ }^{4} C_{2} \times{ }^{12} C_{1} \times{ }^{4} C_{3}}{{ }^{52} C_{5}} $
$= \frac{13 \times 6 \times 12 \times 4}{\frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1}} $
$= \frac{13 \times 6 \times 12 \times 4}{52 \times 51 \times 10 \times 49 \times 2} $
$= \frac{6}{4165}$