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: Determine if order matters

When drawing a king and a queen, the order (king then queen or queen then king) doesn't matter for the combination of having one king and one queen. So we use combinations (${}_nC_r$) instead of permutations (${}_nP_r$).

Step2: Calculate number of ways to choose a king

There are 4 kings, and we want to choose 1. The number of ways is ${}_4C_1$.

Step3: Calculate number of ways to choose a queen

There are 4 queens, and we want to choose 1. The number of ways is ${}_4C_1$.

Step4: Calculate total number of ways to choose 2 cards

There are 52 cards, and we want to choose 2. The number of ways is ${}_{52}C_2$.

Step5: Form the probability expression

The probability is the number of favorable outcomes (choosing 1 king and 1 queen) divided by the number of total outcomes (choosing 2 cards). So the expression is $\frac{({}_4C_1)({}_4C_1)}{{}_{52}C_2}$.

Answer:

$\boldsymbol{\frac{({}_4C_1)({}_4C_1)}{{}_{52}C_2}}$ (the last option)