Understanding the Maximum Number of Students Who Don't Like At Least One Fruit

Given in a class of 25 students, we have the following information about their fruit preferences:

21 students like apples 22 students like oranges 19 students like pears

The key question here is, what is the maximum possible number of students that don't like at least one fruit? To answer this, we need to carefully consider the overlaps in preferences among the students.

Initial Analysis

At first glance, it might seem difficult to determine the exact number of students who don't like at least one fruit. However, we can use set theory and logical reasoning to find the maximum possible number.

Maximum Possible Non-Fruit Lovers

If we assume the worst-case scenario where the maximum number of students might not like at least one fruit, we can use the principle of inclusion-exclusion.

Let's define:

A: Set of students who like apples B: Set of students who like oranges C: Set of students who like pears

The number of students who like at least one fruit can be calculated as follows:

|A ∪ B ∪ C| |A| |B| |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| |A ∩ B ∩ C|

However, without specific information on overlaps, we need to make assumptions. If we consider the worst-case scenario, where the overlaps are minimized, we can assume that the maximum number of non-Fruit lovers might be 3. This is because the minimum number of students who like all three fruits can be calculated to account for any overlaps.

Calculating the Maximum Non-Fruit Lovers

Let's break it down step-by-step:

1. If the students who like apples and pears also like oranges, we have:

25 - 22 3 students who don't like at least one fruit.

2. If we consider the complementary probability:

Probability of no fruit 1 - P(a) - P(o) - P(p) 25 - 21/25 * 25 - 22/25 * 25 - 19/25

This simplifies to:

4/25, 3/25, 6/25

3. The key is the wording "at least one fruit." This means the students could be liking one or more fruits. Therefore, the maximum number of students who don’t like any fruit is 3.

Actual Maximum Calculation

The actual maximum number of students who don't like at least one fruit could indeed be 0 to 3. In the worst-case scenario, only 3 students might not like any fruit. This is because the information provided allows for flexibility in overlaps.

Conclusion

The maximum number of students who don't like at least one fruit is 3. This is based on the assumptions and the worst-case overlap scenario among the students' fruit preferences.

Remember, the key is to carefully analyze the given data and consider the overlaps in preferences to find the maximum non-Fruit lovers.

Key Takeaways

Worst-case overlap scenario leads to a maximum of 3 non-Fruit lovers. The given information allows for varying overlap assumptions. The exact number could range from 0 to 3.

By understanding these principles, you can approach similar set theory problems with confidence.