Question

From a pack of playing cards three cards are drawn simultaneously. The probability that these are one king, one queen and one jack is

• A. $\frac{64}{5525}$
• B. $\frac{16}{5525}$
• C. $\frac{128}{5525}$
• D. $\frac{64}{625}$

Answer 1

The correct option is B

$\frac{16}{5525}$

Explanation for the correct option:

Step: 1 Find the ways a king can be selected from $4$ kings.

There are $4$ kings in that pack.

The different ways a king can be selected from $4$ kings is

$$\begin{gathered} =C_1^4 \\ =4 \end{gathered}$$

Step: 2 Find the ways a queen can be selected from $4$ queen.

There are $4$ queens in that pack.

The different ways a queen can be selected from $4$ queens is

$$\begin{gathered} =C_1^4 \\ =4 \end{gathered}$$

Step: 3 Find the ways a jack can be selected from $4$ jacks.

There are $4$ jacks in that pack.

The different ways a jack can be selected from $4$ jacks is

$$\begin{gathered} =C_1^4 \\ =4 \end{gathered}$$

Step: 4 Find the ways any $3$ cards can be selected from $52$ cards.

There are $52$ cards in that pack.

The different ways any $3$ cards can be selected from $52$ cards is

$$\begin{gathered} =C_3^{52} \\ =\frac{52!}{3!\times 49!} \\ =\frac{52\times 51\times 50\times 49!}{3\times 2\times 49!} \\ =22100 \end{gathered}$$

Step: 5 Calculate the required probability.

The probability that these are one king, one queen and one jack is

=$$\begin{gathered} =\frac{4\times 4\times 4}{22100} \\ =\frac{\mathbf{16}}{\mathbf{5525}} \end{gathered}$$

Hence, Option (B) is the correct answer.