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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Given that, Total number of students n(U)=200Students who study Mathematics n(M)=120Students who study Physics n(P)=90Students who study Chemistry n(C)=70Students who study Mathematics and Physics n(M∩P)=40Students who study Physics and Chemistry (P∩C)=30Students who study Chemistry and Mathematics n(C∩M)=50Students who study none of these subjects n(M∪P∪C)'=20
Now,n(M∪P∪C)'=n(U)−n(M∪P∪C)
⇒20=200−n(M∪P∪C)
⇒n(M∪P∪C)=200−20
⇒n(M∪P∪C)=180
So,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)=20
Hence, 20 students study all the three
subjects
Now,n(M∪P∪C)'=n(U)−n(M∪P∪C)
⇒20=200−n(M∪P∪C)
⇒n(M∪P∪C)=200−20
⇒n(M∪P∪C)=180
So,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)=20
Hence, 20 students study all the three
subjects
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