Final answer:
The largest number of cars with the same color and options that can be guaranteed out of 100,000 is at least 3126, calculated using the Pigeonhole Principle with 32 possible color and option combinations.
Step-by-step explanation:
The question involves finding the largest number of cars that can have the same color and the same options out of 100,000 cars sold with given customizations.
With three options available for each car, each of which can be either selected or not, there are 23 possible combinations of options since each option has two possible states: taken or not taken. With four colors available, this creates a total of 23 × 4 = 32 different combinations for options and color.
Using the Pigeonhole Principle, to guarantee that at least two cars have the same combination of color and options, you would divide the total number of cars sold by the number of possible combinations and round down to the nearest whole number.
Therefore, 100,000 cars divided by 32 possible combinations equals 3125 whole combinations with at least one car leftover, meaning at least 3126 of these cars will share the same color and options configuration.
This is the largest number we can guarantee will have the same color and options out of the 100,000 cars sold. It is important to note that in real scenarios such as purchasing a car, factors like imperfect information about the condition of used cars and the associated risks are important considerations.
Marvin's concerns in the provided text about the potential of acquiring a 'lemon,' represent the challenges in ensuring high-quality purchases and the balance between cost and reliability.