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Try this problem from I.S.I. B.Stat Entrance Objective Problem from TOMATO based on Sets and Probability. You may use sequential hints to solve the problem.

Sixty students appeared in a test consisting of three papers I ,II, and III. Of these students, 25 passed in paper I, 20 in paper II and 8 in paper III. Further 42 students passed in at least one of papers I and II, 30 in at least one of papers I and III, 25 in at least one of papers II and III. Only one student passed in all the three papers. Then the number of students who failed in all the three papers is

- 17
- 15
- 45
- 33

Sets

Probability

Algebra

But try the problem first...

Answer: 15

Source

Suggested Reading

B.Stat Objective Problem

Challenges and Thrills of Pre-College Mathematics by University Press

First hint

Let failed P I=P(A)=35, p II=P(B)=40, p III =P(C)=52, P (IandII)=\(P(A \bigcap B)\)= 18 P (I and III)=\(P(A \bigcap C)\)=30 P (II and III)=\(P(B \bigcap C)\)=35 P (I or II or III)=\(P(A \bigcup B \bigcup C)\)=59

Second Hint

Then by Poincare Theorem, \(P(A \bigcup B \bigcup C)\)=P(A)+P(B)+P(C)-\(P(A \bigcap B)\)-\(P(A \bigcap C)\)-\(P(B \bigcap C)\)+x

Then 59=35+40+52-18-30-35+x where x is the required value

Final Step

then x=59-44=15.

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