In a car agency one day 120 cars were decorated with three different accessories viz., power window, AC and music system. 80 cars were decorated wtih power windows, 80 cars were decorated with AC and 80 cars were decorated with music systems. What is the minimum and maximum number of cars which were decorated with all of three accessories?

Options

A. 10,61
B. 10,45
C. 25,35
D. None of these

Answer

To solve the problem step by step, we will analyze the given information and use set theory concepts to find the minimum and maximum number of cars decorated with all three accessories: power windows, AC, and music systems.

### Step 1: Understand the Given Information
We have:
- Total cars = 120
- Cars with power windows = 80
- Cars with AC = 80
- Cars with music systems = 80

### Step 2: Define Variables
Let:
- \( X \) = Number of cars decorated with all three accessories (power windows, AC, and music system).

### Step 3: Calculate the Minimum Number of Cars with All Three Accessories
From the information given, we can derive the following:
- Cars with only power windows = \( 80 - X \)
- Cars with only AC = \( 80 - X \)
- Cars with only music systems = \( 80 - X \)

Now, if we assume that the maximum overlap occurs (i.e., the minimum number of cars that do not have all three accessories), we can calculate the total number of cars decorated with at least one of the accessories.

If we consider the worst-case scenario where the cars are decorated with only two accessories, we can calculate:
- The total number of cars decorated with at least one accessory can be calculated as:
\[ \text{Total cars with at least one accessory} = \text{Cars with power windows} + \text{Cars with AC} + \text{Cars with music systems} - 2X \]
This is because the cars with all three accessories are counted three times (once in each category), so we subtract them twice.

Substituting the numbers:
\[ \text{Total cars with at least one accessory} = 80 + 80 + 80 - 2X = 240 - 2X \]

### Step 4: Set Up the Inequality
Since the total number of cars is 120, we can set up the inequality:
\[ 240 - 2X \leq 120 \]
Solving for \( X \):
\[ 240 - 120 \leq 2X \]
\[ 120 \leq 2X \]
\[ X \geq 60 \]

### Step 5: Calculate the Maximum Number of Cars with All Three Accessories
For the maximum number of cars decorated with all three accessories, we consider the scenario where the overlap is maximized. The maximum occurs when all cars with power windows, AC, and music systems are the same cars.

Since there are 120 cars in total and each category has 80 cars, the maximum number of cars that can have all three accessories cannot exceed the smallest group:
\[ X \leq 80 \]

### Conclusion
Thus, the minimum number of cars decorated with all three accessories is 60, and the maximum number is 80.

### Final Answer
- Minimum number of cars decorated with all three accessories: **60**
- Maximum number of cars decorated with all three accessories: **80**