Question

the smith family is shopping for a new car and they are basing their decision on color and style. explain how many color choices and how many style choices they might have if there are 8 possible outcomes. justify your answer.

Explanation:

Step1: Recall the fundamental counting principle

The fundamental counting principle states that if there are \( m \) ways to choose a color and \( n \) ways to choose a style, the total number of outcomes is \( m\times n \). Here, the total number of outcomes is 8, so \( m\times n = 8 \).

Step2: Find factor pairs of 8

We find all pairs of positive integers \((m, n)\) such that \( m\times n = 8 \). The factor pairs of 8 are (1, 8), (2, 4), (4, 2), and (8, 1).

Step3: Interpret the factor pairs

Each factor pair represents a possible number of color choices (\( m \)) and style choices (\( n \)). For example, if \( m = 2 \) (2 color choices), then \( n = 4 \) (4 style choices) because \( 2\times4 = 8 \). Similarly, if \( m = 4 \), then \( n = 2 \), and so on. This shows the possible combinations of color and style choices that result in 8 total outcomes.

Answer:

To determine the number of color and style choices, we assume the number of color choices is \( c \) and the number of style choices is \( s \). The total number of outcomes is the product of the number of color choices and style choices, so \( c\times s = 8 \).

We find the factor pairs of 8:

• If \( c = 1 \), then \( s = \frac{8}{1}=8 \)
• If \( c = 2 \), then \( s = \frac{8}{2}=4 \)
• If \( c = 4 \), then \( s = \frac{8}{4}=2 \)
• If \( c = 8 \), then \( s = \frac{8}{8}=1 \)

So possible pairs of (color choices, style choices) are (1, 8), (2, 4), (4, 2), (8, 1). For example, if there are 2 color choices, there would be 4 style choices (since \( 2\times4 = 8 \)), or if there are 4 color choices, there would be 2 style choices, etc. The key is that the product of the number of color and style choices equals the total number of outcomes (8) by the fundamental counting principle, which states that if there are \( m \) ways to do one thing and \( n \) ways to do another, there are \( m\times n \) ways to do both.