Suppose that 100 of 120 students at a college take at least one of the languages France, Germany, and Russia also suppose 65 study France, 45 study Germany ,42 study Russia, 20 study France and Germany, 25 study France and Russia and 15 study Germany and Russia. Find number of students (a) who study all three language (b) who study only Russia (c)who study France or Germany.
Answers
Given : 100 of 120 students at a college take at least one of the languages France, Germany, and Russia
65 study France, 45 study Germany ,42 study Russia,
20 study France and Germany,
25 study France and Russia
15 study Germany and Russia.
To Find : number of students
(a) who study all three language
(b) who study only Russia
(c)who study France or Germany
Solution:
100 of 120 students at a college take at least one of the languages France, Germany, and Russia
F ∪ G ∪ R = 100
F ∪ G ∪ R = F + G + R - F ∩ G - R ∩ G - F ∩ R + F ∩ G ∩ R
=> 100 = 65 + 45 + 42 - 20 - 15 - 25 + F ∩ G ∩ R
=> 100 = 92 + F ∩ G ∩ R
=> F ∩ G ∩ R = 8
8 study all three language
study only Russia = R - R ∩ G - F ∩ R + F ∩ G ∩ R
= 42 - 15 - 25 + 8
= 10
study France or Germany = F + G - F ∩ G
= 65 + 45 - 20
= 90
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