In a class of 200 students,70 offered physics,90 chemistry,100 mathematics while 24 did not offer any of the three subject,23 student offered physics and chemistry,41 chemistry and mathematics while 8 offered all the three subjects.How many students offered exactly two of the subjects?
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Let P be the number of students offered Physics (P=70),
C be the number of students offered Chemistry (C=90),
M be the number of students offered Mathematics (M=100),
PC be the number of students offered Physics and Chemistry (PC=23),
CM be the number of students offered Chemistry and Mathematics (CM = 41),
PM be the number of students offered Physics and Mathematics (PM = ?),
PCM be the number of students offered Physics, Chemistry and Mathematics (PCM = 8),
P|C|M be the number of students offered at least one of three subjects.
P|C|M = 200 - 24 = 176.
Using the inclusion-exclusion principle, we have:
P|C|M = P + C + M - PC - PM - CM +PCM,
from where
PM = P + C + M - PC - CM +PCM - P|C|M = 70 + 90 + 100 - 23 - 41 + 8 - 176 = 28.
Let now be the number of students offered Physics and Chemistry but not Mathematics, be the number of students offered Physics and Mathematics but not Chemistry, and be the number of students offered Mathematics and Chemistry but not Physics. Then
The total number of students offered exactly 2 subjects is equal to
Answer. 68 students offered exactly 2 subjects.