Math Problem Statement

A person can order a new car with a choice of

1212

possible​ colors, with or without air​ conditioning, with or without automatic​ transmission, with or without power​ windows, and with or without a CD player. In how many different ways can a new car be ordered with regard to these​ options?  

Solution

To determine the total number of ways a new car can be ordered with regard to these options, we need to account for the number of choices for each feature.

Step 1: Analyze the options

  1. Color: 1212 choices.
  2. Air conditioning: 2 choices (with or without).
  3. Automatic transmission: 2 choices (with or without).
  4. Power windows: 2 choices (with or without).
  5. CD player: 2 choices (with or without).

Step 2: Multiply the choices

The total number of combinations is given by multiplying the choices for each feature:

Total combinations=(choices for color)×(choices for air conditioning)×(choices for automatic transmission)×(choices for power windows)×(choices for CD player)\text{Total combinations} = (\text{choices for color}) \times (\text{choices for air conditioning}) \times (\text{choices for automatic transmission}) \times (\text{choices for power windows}) \times (\text{choices for CD player})

Substitute the values:

Total combinations=1212×2×2×2×2\text{Total combinations} = 1212 \times 2 \times 2 \times 2 \times 2

Total combinations=1212×16=19,392\text{Total combinations} = 1212 \times 16 = 19,392

Final Answer:

The car can be ordered in 19,392 different ways.


Would you like me to provide additional details or a breakdown of this solution?

Here are 5 related questions:

  1. What if the car manufacturer offered more or fewer colors? How would that change the calculation?
  2. How would the calculation change if additional features were added?
  3. What if some combinations (e.g., certain colors with specific options) were restricted?
  4. Can this method be generalized to other customizable products?
  5. How can we represent this problem using permutations or combinations?

Tip: When solving similar problems, clearly list all features and their options to avoid missing any combinations.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Multiplication Principle

Formulas

Total combinations = (choices for color) × (choices for air conditioning) × (choices for automatic transmission) × (choices for power windows) × (choices for CD player)

Theorems

Multiplication Principle of Counting

Suitable Grade Level

Grades 8-10