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

Introduction / Context:
This problem uses basic set theory and the principle of inclusion and exclusion. Students can take French, German, both or neither. You are asked to find how many students are not enrolled in either course, given the numbers taking each subject and the overlap between them.

Given Data / Assumptions:

• Total number of students = 78.

• Number of students taking French = 41.

• Number of students taking German = 22.

• Number of students taking both French and German = 9.

• We assume each student is correctly counted in these figures and can belong to zero, one or both course groups.

Concept / Approach:
To find how many students take at least one of the two subjects, we use inclusion and exclusion: |French ∪ German| = |French| + |German| − |both|. Once we know how many students are in either or both courses, we subtract that count from the total number of students to find those who take neither course. This approach avoids double counting students who are in both groups.

Step-by-Step Solution:
Number taking French = 41. Number taking German = 22. Number taking both = 9. Number taking French or German (union) = 41 + 22 − 9. Compute: 41 + 22 = 63, 63 − 9 = 54. So, 54 students are enrolled in at least one of the two courses. Total students = 78. Number of students not enrolled in either course = 78 − 54 = 24.

Verification / Alternative check:
As a check, imagine dividing the class into four groups: French only, German only, both and neither. We know that French only + both = 41, German only + both = 22 and both = 9. Therefore French only = 41 − 9 = 32, German only = 22 − 9 = 13. Now total in at least one course = French only + German only + both = 32 + 13 + 9 = 54. Subtracting from 78 again gives 78 − 54 = 24 students taking neither course. The consistency of this count confirms the answer.

Why Other Options Are Wrong:
Answers such as 6, 12 or 18 would imply different union sizes that do not align with the given counts. For example, if 12 students took neither, then 78 − 12 = 66 students would be taking French or German, which contradicts the union calculation of 54. Therefore these values are inconsistent with the inclusion and exclusion formula applied to the given data.

Common Pitfalls:
A frequent mistake is to add 41 and 22 without subtracting the 9 students counted twice, which would double count the students taking both courses. Another error is subtracting the overlap twice. To avoid such errors, always remember the union formula: size of union = sum of individual sizes minus the intersection. Carefully substituting values will yield the correct result.

Final Answer:
The number of students not enrolled in either French or German is 24.