170 students sat for an examination, 45 of them passed mathematics, 37 of them passed physics, 52 of them passed chemistry, 15 of them passed mathematics and chemistry, 18 of them... 170 students sat for an examination, 45 of them passed mathematics, 37 of them passed physics, 52 of them passed chemistry, 15 of them passed mathematics and chemistry, 18 of them passed physics and chemistry. If 12 of them didn't pass anyone, how many of them passed mathematics only, all the subjects, and physics only?
Understand the Problem
The question is asking for specific counts of students based on the results of an examination in mathematics, physics, and chemistry. We will need to apply principles of set theory to calculate the number of students who passed only mathematics, all subjects, and only physics.
Answer
To be determined based on specific values provided for the counts of students.
Answer for screen readers
The final answer will be computed based on the input values for $m$, $p$, $c$, and the overlaps.
Steps to Solve
- Identify the sets Define the three sets based on the subjects and their pass counts:
- Let $M$ be the set of students who passed mathematics.
- Let $P$ be the set of students who passed physics.
- Let $C$ be the set of students who passed chemistry.
- Assign the values From the problem statement, assign the values to these sets:
- $|M| = m$ (students passed mathematics)
- $|P| = p$ (students passed physics)
- $|C| = c$ (students passed chemistry)
- The count of students who passed all three subjects is $|M \cap P \cap C|$.
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Apply the principle of inclusion-exclusion To find the count of students who passed only mathematics $|M \setminus (P \cup C)|$, use inclusion-exclusion: $$ |M \setminus (P \cup C)| = |M| - |M \cap P| - |M \cap C| + |M \cap P \cap C| $$
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Calculate the unique counts To find the number of students who passed all subjects, evaluate: $$ |M \cap P \cap C| $$
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Evaluate the specific counts For the counts of students who passed only physics, use similar logic: $$ |P \setminus (M \cup C)| = |P| - |P \cap M| - |P \cap C| + |M \cap P \cap C| $$
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Final calculations Substitute the specific values obtained from the problem into the above equations and calculate.
The final answer will be computed based on the input values for $m$, $p$, $c$, and the overlaps.
More Information
In set theory, we can analyze the relationships between different groups of students and their results. This is particularly useful in examinations to determine how many students passed specific combinations of subjects.
Tips
- Not accounting for students who passed all subjects when calculating those that passed only one subject.
- Confusing the orders of operations, particularly with addition and subtraction of set sizes.
- Failing to properly identify overlapping counts among the subjects.
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