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
$=\frac{{}^{13}{C}_1\times {}^4{C}_2\times {}^{12}{C}_1\times {}^4{C}_3}{{}^{52}{C}_5}$
$=\frac{13\times 6\times 12\times 4}{\frac{52\times 51\times 50\times 49\times 48}{5\times 4\times 3\times 2\times 1}}$
$=\frac{13\times 6\times 12\times 4}{52\times 51\times 10\times 49\times 2}$
$=\frac{6}{4165}$