In a class of 78 students, 41 are taking French and 22 are taking German. Of the students taking French or German, 9 are taking both courses. How many students are not enrolled in either French or German?

Introduction / Context:
This question is based on set theory and the principle of inclusion and exclusion. It uses a very common exam pattern where students are divided according to interests in two subjects. The goal is to find how many students are outside both groups. Understanding how to handle overlaps between sets is important for many aptitude and data interpretation problems.

Given Data / Assumptions:

Total number of students in the class is 78.
41 students are taking French.
22 students are taking German.
9 students are taking both French and German.
We need the count of students who are not enrolled in either French or German.

Concept / Approach:
For two sets A and B, the number of elements in A union B is given by n(A ∪ B) = n(A) + n(B) - n(A ∩ B). Here, A is the set of students taking French and B is the set of students taking German. After finding how many students are taking at least one of the two languages, we subtract this count from the total number of students to get those who take neither language.

Step-by-Step Solution:
Step 1: Identify the sets. Let A be the set of students taking French and B be the set of students taking German. Step 2: Use the inclusion and exclusion principle. Number taking French or German = n(A ∪ B) = n(A) + n(B) - n(A ∩ B). Step 3: Substitute the given values. n(A ∪ B) = 41 + 22 - 9 = 63 - 9? Wait carefully: 41 + 22 = 63, and 63 - 9 = 54. So 54 students are taking at least one of the two languages. Step 4: Compute those taking neither language. Students taking neither French nor German = total students - students taking at least one language. So required number = 78 - 54 = 24.

Verification / Alternative Check:
We can mentally confirm that 24 students out of 78 not taking any language means 54 are taking at least one. Given 41 take French and 22 take German with 9 in the overlap, French only is 41 - 9 = 32, German only is 22 - 9 = 13, and both is 9. Total who take at least one language is 32 + 13 + 9 = 54, which matches our earlier calculation, so the answer is consistent.

Why Other Options Are Wrong:
Option 6 is far too small and would mean almost all students are in at least one language, inconsistent with the given numbers.
Option 12 does not match the inclusion and exclusion calculation and would lead to an incorrect total when recomputed.
Option 18 also fails when we reassemble the counts of French only, German only and both; it does not align with the total of 78 students.

Common Pitfalls:
One common mistake is to simply add 41 and 22 and ignore the overlap, which double counts the students taking both languages. Another error is subtracting the intersection twice. Some students also forget that the remainder after counting A ∪ B represents those who are in neither set, which is exactly what the question asks for.

Final Answer:
The number of students not enrolled in either course is 24.