In a school of 250 students, 72 students study mathematics, 65 study Chemistry, 83 study Geography. In addition 56 students study Mathematics and Geography, 47 study Mathematics and Chemistry and 62 study Geography and Chemistry. If 27 students study all the three subjects, how many study: (i) Only one subject? (ii) Exactly two subjects? (iii) None of the three subjects?
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Step-by-step explanation:
In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics and 70 study Chemistry, 40 study Mathematics and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Mathematics and 20 none of these subjects. Find the number of students who study all the three subjects
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M = Mathematics ; P = Physics and C = Chemistry
n(M) = 120 n(P) = 90 n (C) = 70 n ( M ∩ P) = 40
n ( P ∩ C ) = 30 n ( C ∩ M ) = 50 n ( M ∪ P ∪ C )’ = 20
Now n(M ∪ P ∪ C)’ = n(U) – n(M ∪ P ∪ C)
20 = 200 – n (M ∪ P ∪ C)
Therefore, n(M ∪ P ∪ C) = 200 – 20 = 180
n(M ∪ P ∪ C)
= n(M) + n(P) + n(C) – n(M ∩ P) – n(P ∩ C) – n(C ∩ M) + n(M ∩ P ∩ C)
180 = 120 + 90 + 70 - 40 - 30 - 50 + n(M ∩ P ∩ C)
⇒ n(M ∩ P ∩ C) =180 - 120 - 90 - 70 + 40 + 30 + 50
⇒ n(M ∩ P ∩ C) = 20.
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