Solution to In a class of 35 students it is known that 24 of them do arts, … - Sikademy
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Archangel Macsika

In a class of 35 students it is known that 24 of them do arts, 20 do chemistry and 22 do biology. All the students do at least one of the 3 subjects, 3 do all the three subjects while 7 do art and biology. 6 do art and chemistry but not biology and 8 do chemistry and biology. How many of them do chemistry or biology only or arts only?

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In a class of 45 students 


N(A\cup C\cup B)=45

It is known that 24 of them do arts, 20 do chemistry and 22 do biology. 

N(A)=24, N(C)=20, N(B)=22

3 do all the three subjects


N(A\cap C\cap B)=3

7 do art and biology


N(A\cap B)=7

6 do art and chemistry but not biology


N(A\cap C \cap B^C)=6

8 do chemistry and biology

N(C\cap B)=8

Then

N(A\cap B \cap C^C)=N(A\cap B)-N(A\cap B\cap C)

=7-3=4

N(B\cap C\cap A^C)=N(B\cap C)-N(A\cap B\cap C)

8-3=5




N(A\cap B^C\cap C^C)=N(A)-N(A\cap B\cap C^C)

-N(A\cap C\cap B^C)-N(A\cap B\cap C)

=24-4-6-3=11


N(B\cap A^C\cap C^C)=N(B)-N(B\cap A\cap C^C)

-N(B\cap C\cap A^C)-N(A\cap B\cap C)

=22-4-5-3=10


N(C\cap A^C\cap B^C)=N(C)-N(C\cap A\cap B^C)

-N(C\cap B\cap A^C)-N(A\cap B\cap C)

=20-6-5-3=6

Check


11+10+6+4+6+5+3=45



N(\text{chemistry or biology only})=N(C\cap B\cap A^C)=5


N(\text{arts only })=N(A\cap B^C\cap C^C)=11


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