MathLover1121
Answered

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In a school of 100 students, 90 study English, 75 study Spanish and 42 study French. Every student must study at least one of the three languages. What is the least possible number of students who could be studying all three languages?

Sagot :

The number of students that study each language can be expressed by

using the equation for the union of three sets.

  • The least possible number of student that study all three languages is zero.

Reasons:

The number of students in the school, U = 100

Number of students that study English, n(A) = 90

Number of students that study Spanish, n(B) = 75

Number of students that study French, n(C) = 42

Number of languages each student must study ≥ 1

Required:

The least possible number of students that study all three languages.

Solution;

Number of students that study all three languages = n(A∩B∩C)

U = n(A∪B∪C)

From set theory, equation for three sets, we have;

  • n(A∩B∩C) = n(A∪B∪C) - (n(A) + n(B) + n(C) - n(A∩B) - n(A∩C) - n(B∩C))  

Therefore;

n(A∩B∩C) = 100 - (90 + 75 + 42 - n(A∩B) - n(A∩C) - n(B∩C))

The least value of n(A∩B∩C) = 0

Therefore;

100 - (90 + 75 + 42 - n(A∩B) - n(A∩C) - n(B∩C))  = 0

100 = (207 - n(A∩B) - n(A∩C) - n(B∩C))

n(A∩B) + n(A∩C) + n(B∩C) = 207 - 100 = 107

n(A∩B) + n(A∩C) + n(B∩C) = 107

The above equation is possible

Therefore;

n(A∩B∩C) = 100 - (90 + 75 + 42 - 107) = 0

  • The least possible number of students that study all three languages, n(A∩B∩C) is 0

Learn more about set theory here:

https://brainly.com/question/13968184

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