Let's solve this problem step by step:
1. Identify the Problem:
We are asked to determine the probability of drawing one king and one queen from a standard deck of 52 cards.
2. Total Possible Combinations:
We need to find the total number of ways to choose 2 cards out of the 52-card deck. This can be calculated using combinations since the order of selecting the cards does not matter.
[tex]\[
\binom{52}{2}
\][/tex]
From the provided answer, we know:
[tex]\[
\binom{52}{2} = 1326
\][/tex]
3. Number of Ways to Choose 1 King:
There are 4 kings in the deck, and we need to choose 1 of them. This is again a combination:
[tex]\[
\binom{4}{1}
\][/tex]
From the provided answer, we know:
[tex]\[
\binom{4}{1} = 4
\][/tex]
4. Number of Ways to Choose 1 Queen:
Similarly, there are 4 queens in the deck, and we need to choose 1 of them.
[tex]\[
\binom{4}{1}
\][/tex]
From the provided answer, we know:
[tex]\[
\binom{4}{1} = 4
\][/tex]
5. Total Number of Favorable Combinations:
To get one king and one queen, we multiply the number of ways to choose one king by the number of ways to choose one queen:
[tex]\[
4 \times 4 = 16
\][/tex]
6. Calculate the Probability:
The probability is the number of favorable outcomes divided by the total number of possible outcomes:
[tex]\[
\frac{16}{1326}
\][/tex]
From the provided answer, we know:
[tex]\[
\frac{16}{1326} \approx 0.012066
\][/tex]
Given options are listed in a problems statement, we can see that:
- The first and third options use permutations, which are incorrect because the order doesn't matter.
- The second option appears to be correct since it uses combinations correctly:
[tex]\[
\frac{\binom{4}{1} \binom{4}{1}}{\binom{52}{2}}
\][/tex]
- The last option is malformed with unstructured dot, so it cannot be correct.
Therefore, the correct expression representing the probability of drawing a king and a queen is:
[tex]\[
\frac{\left({ }_4 C_1\right)\left({ }_4 C_1\right)}{{}_{52} C_2}
\][/tex]
Which confirms that the correct option is:
[tex]\[
\boxed{\frac{\left({ }_4 C_1\right)\left({ }_4 C_1\right)}{{ }_{52} C_2}}
\][/tex]
