To find the probability of drawing a king and a queen from a standard deck of 52 playing cards, follow these steps:
1. Identify the total number of ways to draw 2 cards from the deck.
This can be calculated using combinations since the order does not matter:
[tex]\[
\binom{52}{2} = \frac{52 \times 51}{2 \times 1} = 1326
\][/tex]
2. Identify the number of ways to draw one king and one queen.
We need to choose 1 king out of the 4 kings and 1 queen out of the 4 queens. Again, use combinations:
[tex]\[
\binom{4}{1} = 4 \quad \text{(ways to choose a king)}
\][/tex]
[tex]\[
\binom{4}{1} = 4 \quad \text{(ways to choose a queen)}
\][/tex]
Thus, the total number of ways to draw one king and one queen is:
[tex]\[
4 \times 4 = 16
\][/tex]
3. Calculate the probability.
The probability is the ratio of the number of favorable outcomes to the total number of outcomes:
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{16}{1326}
\][/tex]
From the above steps, the correct expression to represent the probability is:
[tex]\[
\frac{\left. \binom{4}{1}\right)\left(\binom{4}{1}\right)}{\binom{52}{2}}
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[
\frac{\left.{ }_4 C_1\right)\left({ }_4 C_1\right)}{{ }_{52} C_2}
\][/tex]
