Question

a standard deck of 52 playing cards contains four of each numbered card 2 - 10 and four each of aces, kings, queens, and jacks. two cards are chosen from the deck at random. which expression represents the probability of drawing a king and a queen?
\\(\frac{(_{4}p_{1})(_{3}p_{1})}{_{52}p_{2}}\\)
\\(\frac{(_{4}c_{1})(_{3}c_{1})}{_{52}c_{2}}\\)
\\(\frac{(_{4}p_{1})(_{4}p_{1})}{_{52}p_{2}}\\)
\\(\frac{(_{4}c_{1})(_{4}c_{1})}{_{52}c_{2}}\\)

Explanation:

Step1: Calculate number of ways to choose a king and a queen

There are 4 kings and 4 queens. The number of ways to choose 1 king out of 4 is $_{4}C_{1}$, and the number of ways to choose 1 queen out of 4 is $_{4}C_{1}$. By the multiplication - principle, the number of ways to choose a king and a queen is $(_{4}C_{1})(_{4}C_{1})$.

Step2: Calculate total number of ways to choose 2 cards from 52

The total number of ways to choose 2 cards from a deck of 52 cards is $_{52}C_{2}$.

Step3: Calculate probability

The probability of an event is the number of favorable outcomes divided by the number of total outcomes. So the probability of drawing a king and a queen is $\frac{(_{4}C_{1})(_{4}C_{1})}{_{52}C_{2}}$.

Answer:

$\frac{(_{4}C_{1})(_{4}C_{1})}{_{52}C_{2}}$ (corresponding to the fourth option in the multiple - choice list)