Question

adele is buying a sport utility vehicle. she is trying to choose among a variety of options.
brand: brand a or brand b
transmission: 4-speed standard, 5-speed standard, or automatic
bumper: steel, vinyl, or 2x4 boards
roof: hard top or convertible top
adele calculates that she has 2 + 3 + 3 + 2, or a total of 10, possible vehicle choices.
what is adele’s error?
options:

• she should have added only the transmission and bumper choices.
• she should have added only the brand and roof choices.
• she should have multiplied the total of all the choices.
• she should have multiplied only the brand and roof choices.

Explanation:

Step1: Identify the problem type

This is a problem about counting the number of possible vehicle choices, which involves the multiplication principle in combinatorics. The multiplication principle states that if there are \(n_1\) ways to do one thing, \(n_2\) ways to do a second thing, \(n_3\) ways to do a third thing, and so on, then the total number of ways to do all the things together is \(n_1\times n_2\times n_3\times\cdots\).

Step2: Analyze Adele's options

• Brand: There are 2 options (Brand A or Brand B), so \(n_1 = 2\).
• Transmission: There are 3 options (4 - speed standard, 5 - speed standard, or automatic), so \(n_2=3\).
• Bumper: There are 3 options (steel, vinyl, or 2x4 boards), so \(n_3 = 3\).
• Roof: There are 2 options (hard top or convertible top), so \(n_4=2\).

Step3: Calculate the correct total number of choices

Using the multiplication principle, the total number of possible vehicle choices should be \(n_1\times n_2\times n_3\times n_4=2\times3\times3\times2\).

Step4: Analyze Adele's mistake

Adele added the number of options for each category (\(2 + 3+3 + 2\)) instead of multiplying them. So her error was that she should have multiplied the total of all the choices (the number of options for each category) instead of adding them.

Answer:

She should have multiplied the total of all the choices.