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 ____

• A. 18
• B. 20
• C. 24
• D. 32

Hint:

First find how many students like at least one subject using the union formula, then add the students who like neither.

Answer 1

The Correct Option is C

Step 1: Split the group into four non overlapping regions using a Venn diagram idea.
Any student in the group falls into exactly one of four regions: likes only Mathematics, likes only English, likes both Mathematics and English, or likes neither.

Step 2: Find the "only Mathematics" region.
10 students like Mathematics in total, and 4 of them also like English, so the number who like only Mathematics is $10 - 4 = 6$.

Step 3: Find the "only English" region.
12 students like English in total, and 4 of them also like Mathematics, so the number who like only English is $12 - 4 = 8$.

Step 4: State the "both" region and the "neither" region directly.
The number who like both subjects is given as 4, and the number who like neither subject is given as 6.

Step 5: Add all four regions together.
Total students $=$ (only Mathematics) $+$ (only English) $+$ (both) $+$ (neither) $= 6 + 8 + 4 + 6 = 24$.

Step 6: Cross check with the union formula method.
The union of Mathematics and English lovers is $6 + 8 + 4 = 18$, which matches $n(M) + n(E) - n(M \cap E) = 10 + 12 - 4 = 18$ found by the other method, and adding the 6 who like neither again gives $18 + 6 = 24$, the same final answer.

\[ \boxed{24} \]