Question

in a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. it may be possible that some students do not like any of these three drinks. then the difference between the maximum and minimum possible number of students who like all the three drinks is

Answer 1

The difference between the maximum and minimum possible number of students who like all three drinks (coffee, tea, and lemonade) in a class of 100 students is 1.

Step-by-step explanation:

The question asks to determine the difference between the maximum and minimum possible number of students who like all three drinks: coffee, tea, and lemonade. First, we find the minimum number by using the principle of inclusion-exclusion.


The sum of students liking at least one drink is 73 (coffee) + 80 (tea) + 52 (lemonade). Now, this count can include students more than once if they like multiple drinks, and the total cannot exceed the class size, which is 100. The maximum possible overlap, when everyone likes at least two drinks, would be when each of the students who like coffee and lemonade also like tea, which is the most-liked. Thus, the minimum possible number of students liking all three, when we maximize the overlap, could be 73 (assuming all coffee lovers also like tea and lemonade).

For the maximum number liking all three, we consider that each of three groups must intersect as little as possible. Since 73 + 80 + 52 = 205 and there are only 100 students, at least 205 - 100 = 105 students must be counted in more than one category. If we assume that these 105 students are exactly the number who like two drinks, this leaves us with 100 + 3 - 205 = 0 students who like all three, which is impossible.



We must have some overlap; therefore, our assumption is erroneous. The maximum overlap occurs when we only count the two largest groups (73 coffee + 80 tea = 153), which means at least 153 - 100 = 53 students like both coffee and tea. So the maximum number who like all three drinks cannot exceed the number who like lemonade, which is 52. With 53 needing to like both coffee and tea, we can say that at least 1 must also like lemonade, making 1 the maximum number for those who like all three.

Therefore, the maximum number of students who like all three drinks is 1 and the minimum is 0, leading to a difference of 1 - 0 = 1.