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 (e.g. $2$ 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

The correct option is A

$\frac{6}{4165}$

Explanation for the correct option:

Find the probability:

Total number of cards in a deck $=52$

Number of favorable cases $={}^{13}C_1\times {}^{4}C_2\times {}^{12}C_1\times {}^{4}C_3$

Now, Probability of drawing 5 cards at random $={}^{52}C_5$

$\therefore$The required probability $=\frac{{}^{13}C_1\times {}^{4}C_2\times {}^{12}C_1\times {}^{4}C_3}{{}^{52}C_5}$

$$\begin{gathered} =\frac{13\times 6\times 2\times 4}{52\times 51\times 10\times 49\times 2} \\ =\frac{\mathbf{6}}{\mathbf{4165}} \end{gathered}$$ $\left[\mathbf{\because }{}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{!}\mathbf{r}\mathbf{!}}\right]$

Hence, Option ‘A’ is Correct.