To analyze the performance of the students in the examination, let’s define some variables based on the information provided:
- Let \( F \) be the set of students who passed French.
- Let \( M \) be the set of students who passed Mathematics.
- Let \( |F| \) be the number of students who passed French.
- Let \( |M| \) be the number of students who passed Mathematics.
- Let \( |F \cap M| \) be the number of students who passed both subjects.
- Let \( |F - M| \) be the number of students who passed only French.
- Let \( |M - F| \) be the number of students who passed only Mathematics.
- Let \( |F \cup M| \) be the number of students who passed at least one subject.
We are given the following data:
- \( |F - M| = 14 \) (students who passed only French)
- \( |F \cap M| = 23 \) (students who passed both subjects)
- \( |F \cup M| = |F| + |M| - |F \cap M| \) (the principle of inclusion-exclusion)
- The total number of students \( = 50 \)
- The number of students who failed both subjects \( = 5 \)
From this, we can determine the number of students who passed at least one subject: \[ |F \cup M| = 50 - 5 = 45 \]
Now, we can express the total number of students who passed: \[ |F \cup M| = |F - M| + |M - F| + |F \cap M| \] Substituting the given values: \[ 45 = 14 + |M - F| + 23 \] Simplifying this gives: \[ 45 = 37 + |M - F| \] Therefore: \[ |M - F| = 45 - 37 = 8 \]
Now, we can summarize:
- The number of students who passed only French (\( |F - M| \)) is 14.
- The number of students who passed only Mathematics (\( |M - F| \)) is 8.
- The number of students who passed both subjects (\( |F \cap M| \)) is 23.
- The total number of students who passed exactly one subject is given by: \[ \text{Exact one subject} = |F - M| + |M - F| = 14 + 8 = 22 \]
To summarize:
- The number of students who passed in French: 14 (only French) + 23 (both) = 37.
- The number of students who passed in Mathematics only: 8.
- The number of students who passed in exactly one subject: 22.
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