Question
here are 100 students in a class. In an examination, 50 of them
ailed in Mathematics, 45 failed in Physics, 40 failed in Biology
nd 32 failed in exactly two of three subjects. Only one studen
assed in all the subjects. Then the number of students failing
all the three subjects
a) Is 12
a) Is 4
t) Is 2
d) Cannot be determined from the given information
ailed in Mathematics, 45 failed in Physics, 40 failed in Biology
nd 32 failed in exactly two of three subjects. Only one studen
assed in all the subjects. Then the number of students failing
all the three subjects
a) Is 12
a) Is 4
t) Is 2
d) Cannot be determined from the given information
Ask by Hart Luna. in Nigeria
Dec 28,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
2 students failed all three subjects.
Solution
Let’s analyze the problem step by step.
Given:
- Total students = 100
- Failed in Mathematics (M) = 50
- Failed in Physics § = 45
- Failed in Biology (B) = 40
- Failed in exactly two subjects = 32
- Only one student passed all subjects (i.e., 99 failed at least one subject)
Let:
-
= Number of students who failed all three subjects.
Using the principle of inclusion-exclusion:
Where:
-
, , = Students failing exactly two subjects. -
(students failing all three).
Given that:
Plugging in the values:
Answer:
2
2
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Bonus Knowledge
To solve this problem, we can use the principle of inclusion-exclusion and some logical deduction. Let’s define the sets:
- Let M be the set of students who failed Mathematics.
- Let P be the set of students who failed Physics.
- Let B be the set of students who failed Biology.
We know:
- |M| = 50
- |P| = 45
- |B| = 40
- |M ∩ P ∩ B| = x (this is the number of students who failed all three subjects)
- 32 students failed in exactly two subjects.
We can denote the number of students who failed exactly two subjects as follows:
- |M ∩ P| - x (failed Math and Physics but not Biology)
- |M ∩ B| - x (failed Math and Biology but not Physics)
- |P ∩ B| - x (failed Physics and Biology but not Math)
From the information given, the total number of students who failed at least one subject can be calculated as:
N = (students failing only in 1 or 3 subjects) + (students failing exactly 2 subjects) + (students who passed all subjects)
N = (students failing only in 1 or 3 subjects) + (students failing exactly 2 subjects) + (students who passed all subjects)
Given that one student passed all subjects, we obtain the total number of students failing at least one subject and set up the equations:
The total should equal 100; therefore,
50 + 45 + 40 - (students failing exactly 2 only) - 3x (the ones failing all three subjects) = 100 (to avoid double-counting the overlaps).
The total should equal 100; therefore,
50 + 45 + 40 - (students failing exactly 2 only) - 3x (the ones failing all three subjects) = 100 (to avoid double-counting the overlaps).
Finding x using the remaining data leads to the conclusion. Eventually, by carefully piecing together this information, we can determine that the number of students failing in all three subjects is 4.
Furthermore, in situations involving numerous conditions like these, it’s essential to carefully track what is known and what deductions can be made from the overlaps.
Always keep a sharp eye on how each piece of data affects others! It’s not uncommon to become tangled in the various conditions, so organizing information through diagrams or tables can often clarify relationships and help solve the problem more effectively.
