Question

texes mathematics 4-8 (115)

1. an experiment requires drawing three cards from an average deck of 52 playing cards. if the cards are drawn without replacement, how many outcomes are in its sample space?

note: each event is when a card is drawn. since each card is unique, the order of the cards drawn will matter which means this is a basic permutation.

a. 140,608
b. 132,600
c. 156
d. 153

Explanation:

🆕 New Concept Discovered: Permutations and Combinations
Arranging items where order matters.

Step 1: Identify the type of counting problem

The problem asks for the number of outcomes in the sample space when drawing three cards from a standard deck of 52 cards without replacement.

The note states: "Each event is when a card is drawn. Since each card is unique, the order of the cards drawn will matter which means this is a basic permutation."

Because order matters and we are drawing without replacement, we use permutations.

Step 2: Apply the permutation formula

To find the number of ways to arrange \( r \) items chosen from a set of \( n \) distinct items, we use the permutation formula:

$$ P(n, r) = \frac{n!}{(n-r)!} $$

Here, the total number of cards is \( n = 52 \), and we are drawing \( r = 3 \) cards:

$$ P(52, 3) = 52 \times 51 \times 50 $$

Step 3: Calculate the product

Calculate the product of the three choices:

$$ 52 \times 51 = 2652 $$

$$ 2652 \times 50 = 132,600 $$

Answer:

B. 132,600