Minimum Students Studying All Three Languages: A Problem and Its Solution
In a school with 100 students, 60 will study French, 60 will study German, and 60 will study Spanish. The question arises as to what is the minimum number of students who will study all three languages. This article will explore how to solve this problem using the principle of inclusion-exclusion, providing a detailed explanation of the steps involved.
Introduction to the Problem
The task is to determine the smallest number of students x who will study all three languages: French, German, and Spanish, given that 60 students will study each of these languages in a school with a total of 100 students. This problem is a classic application of the principle of inclusion-exclusion in set theory.
The Principle of Inclusion-Exclusion
The principle of inclusion-exclusion for three sets is a fundamental concept in combinatorics. It provides a method to count the number of elements in the union of multiple sets, ensuring that elements that belong to more than one set are only counted once. The formula is as follows:
F ∪ G ∪ S F G S - F ∩ G - G ∩ S - S ∩ F F ∩ G ∩ S
Where:
F: Number of students studying French. G: Number of students studying German. S: Number of students studying Spanish. F ∩ G, G ∩ S, S ∩ F: Number of students studying exactly two languages. F ∩ G ∩ S: Number of students studying all three languages.Setting Up the Problem
Given:
n(F) 60 n(G) 60 n(S) 60 Total number of students 100The total number of students studying at least one language should be less than or equal to 100:
100 ≥ 60 60 60 - n(F ∩ G) - n(G ∩ S) - n(S ∩ F) n(F ∩ G ∩ S)
Simplifying the inequality:
100 ≥ 180 - n(F ∩ G) - n(G ∩ S) - n(S ∩ F) n(F ∩ G ∩ S)
Rearranging gives:
n(F ∩ G) n(G ∩ S) n(S ∩ F) - n(F ∩ G ∩ S) ≥ 80
Indicators and Variables
Define:
a F ∩ G b G ∩ S c S ∩ F x F ∩ G ∩ SThe inequality can be rewritten as:
a b c - x ≥ 80
Maximizing Overlaps
The maximum possible values for a, b, and c are constrained by the total number of students. The maximum number of students studying any two languages is 60. Therefore:
a b c ≤ 60 60 60 - 2x
Substituting this into the inequality:
60 60 60 - 2x - x ≥ 80
Simplifying:
180 - 3x ≥ 80
3x ≤ 100
x ≤ 100/3 ≈ 33.33
Since x must be a whole number, the maximum possible value for x is 33. However, we want the minimum number of students studying all three languages.
Minimizing the Number of Students Studying All Three Languages
To minimize
Assuming no students study more than two languages, the maximum number of students studying any two languages is 60. Therefore:
a b c - x ≥ 80
The minimum x is the number of students that must be counted multiple times to satisfy all conditions:
The minimum x must satisfy:
180 - a - b - c - x 100
This implies:
a b c - x 80
Conclusion
The minimum number of students who will study all three languages is 20. This is the smallest value that satisfies the condition that 180 - 3x ≥ 80, ensuring that the total number of students does not exceed 100.
Thus, the minimum number of students who study all three languages is:
20