In a survey among 80 people, 50 people like arrange marriage and 70 people like love marriage. What is the minimum and maximum number of people like both the marriages respectively?

Options

A. 40,45
B. 40,50
C. 30,40
D. can’t be determined

Answer

To solve the problem, we need to determine the minimum and maximum number of people who like both arranged and love marriages based on the given data.

### Step 1: Understand the given information
We have:
- Total number of people surveyed = 80
- Number of people who like arranged marriage (A) = 50
- Number of people who like love marriage (L) = 70

### Step 2: Use the principle of inclusion-exclusion
The principle of inclusion-exclusion helps us find the number of people who like both types of marriages. We denote:
- Let \( x \) be the number of people who like both arranged and love marriages.

According to the principle of inclusion-exclusion:
\[ A + L - x \leq \text{Total number of people} \]
Substituting the values we have:
\[ 50 + 70 - x \leq 80 \]
This simplifies to:
\[ 120 - x \leq 80 \]
Rearranging gives:
\[ x \geq 40 \]
So, the minimum number of people who like both types of marriages is 40.

### Step 3: Determine the maximum number of people who like both marriages
To find the maximum number of people who like both types of marriages, we need to consider the total number of people surveyed:
\[ x \leq \text{Minimum of (A, L)} \]
Here, the maximum number of people who can like both arranged and love marriages cannot exceed the smaller group. Since the smaller group is arranged marriages (50), we have:
\[ x \leq 50 \]

### Conclusion
Thus, the minimum number of people who like both arranged and love marriages is 40, and the maximum is 50.

### Final Answer:
- Minimum number of people who like both marriages = 40
- Maximum number of people who like both marriages = 50

---