The least possible number of students that study all three languages can
be obtained by analogy using the given constraints.
- The least possible number of students who could be studying all three languages is zero.
Reasons:
Number of students in the school, U = 100
Number of students that study English, n(A) = 90
Number of students studying Spanish, n(B) = 75
Number of students studying French, n(C) = 45
Number of languages each student must study = At least 1
Required:
The least number of students that could study all three languages.
Solution:
The number of students studying all three languages are given as follows;
n(A∩B∩C) = n(A∪B∪C) - (n(A) + n(B) + n(C) - (n(A∩B) + n(A∩C) + n(B∩C))
n(A∪B∪C) = U = 100, given that every student studies a language
Therefore;
- n(A∩B∩C) = U - (n(A) + n(B) + n(C) - (n(A∩B) + n(A∩C) + n(B∩C))
When n(A∩B∩C) = 0, we have;
U - (n(A) + n(B) + n(C) - (n(A∩B) + n(A∩C) + n(B∩C)) = 0
U = (n(A) + n(B) + n(C) - (n(A∩B) + n(A∩C) + n(B∩C))
Which gives;
100 = (90 + 75 + 42 - (n(A∩B) + n(A∩C) + n(B∩C))
100 = 207 - (n(A∩B) + n(A∩C) + n(B∩C))
Therefore;
(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;
- The least number of students who could study all three languages, n(A∩B∩C) = 0
Learn more about sets (Venn diagram) here:
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