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
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