Question:
In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is ____
In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is ____
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Hint:
First work out how many students like at least one of the two subjects using the overlap, then remember to add on the students who like neither subject.
First work out how many students like at least one of the two subjects using the overlap, then remember to add on the students who like neither subject.
Updated On: Jul 28, 2026
- 18
- 20
- 24
- 32
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The Correct Option is C
Solution and Explanation
Step 1: Define the sets involved.
Let \(M\) be the set of students who like Mathematics, and \(E\) be the set of students who like English. We are given \(|M| = 10\), \(|E| = 12\), and \(|M \cap E| = 4\), the number of students who like both subjects. We are also told 6 students like neither subject.
Step 2: Recall the formula for the size of a union of two sets.
When counting how many students like Mathematics or English (or both), we cannot just add \(|M|\) and \(|E|\) directly, because the students who like both would then be counted twice. The correct formula subtracts the overlap once:
\[ |M \cup E| = |M| + |E| - |M \cap E| \]
Step 3: Substitute the given numbers.
\[ |M \cup E| = 10 + 12 - 4 = 18 \]
So 18 students like at least one of the two subjects, Mathematics or English or both.
Step 4: Add in the students who like neither subject.
The total group of students is made up of those who like at least one subject, plus those who like neither. Since 6 students fall into the "neither" category and do not overlap with the 18 students already counted:
\[ \text{Total} = |M \cup E| + \text{neither} = 18 + 6 = 24 \]
Step 5: Check why the other options do not fit.
Option (A), 18, is only the count of students who like at least one subject, it forgets to add the 6 students who like neither. Option (B), 20, would come from a wrong union calculation, for instance forgetting to subtract the overlap correctly. Option (D), 32, would come from wrongly adding \(|M| + |E| + \text{neither}\) without subtracting the overlap at all, which double counts the 4 students who like both subjects.
Final Answer:
Adding the 18 students who like at least one subject to the 6 students who like neither gives a total group size of 24. \[ \boxed{24} \]
Let \(M\) be the set of students who like Mathematics, and \(E\) be the set of students who like English. We are given \(|M| = 10\), \(|E| = 12\), and \(|M \cap E| = 4\), the number of students who like both subjects. We are also told 6 students like neither subject.
Step 2: Recall the formula for the size of a union of two sets.
When counting how many students like Mathematics or English (or both), we cannot just add \(|M|\) and \(|E|\) directly, because the students who like both would then be counted twice. The correct formula subtracts the overlap once:
\[ |M \cup E| = |M| + |E| - |M \cap E| \]
Step 3: Substitute the given numbers.
\[ |M \cup E| = 10 + 12 - 4 = 18 \]
So 18 students like at least one of the two subjects, Mathematics or English or both.
Step 4: Add in the students who like neither subject.
The total group of students is made up of those who like at least one subject, plus those who like neither. Since 6 students fall into the "neither" category and do not overlap with the 18 students already counted:
\[ \text{Total} = |M \cup E| + \text{neither} = 18 + 6 = 24 \]
Step 5: Check why the other options do not fit.
Option (A), 18, is only the count of students who like at least one subject, it forgets to add the 6 students who like neither. Option (B), 20, would come from a wrong union calculation, for instance forgetting to subtract the overlap correctly. Option (D), 32, would come from wrongly adding \(|M| + |E| + \text{neither}\) without subtracting the overlap at all, which double counts the 4 students who like both subjects.
Final Answer:
Adding the 18 students who like at least one subject to the 6 students who like neither gives a total group size of 24. \[ \boxed{24} \]
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