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Previous Year Question (PYQs)

A university is offering elective courses in Mathematics, Economics and Sociology. Each of its 100 undergraduate students has to opt for at least one of these electives. Course enrollment data showed that 47 students enrolled for Mathematics, 47 students enrolled for Economics and 57 students enrolled for Sociology. If 7 students enrolled for all three courses, how many students enrolled for exactly one course?




Solution

Venn Diagram Problem: Students in Elective Courses

Given:

  • Total students = 100
  • Math (M) = 47
  • Economics (E) = 47
  • Sociology (S) = 57
  • All three (M ∩ E ∩ S) = 7

Objective:

Find the number of students who enrolled in exactly one course.

Step-by-step:

Let’s use the formula for total union of 3 sets:

Total = M + E + S − (M∩E) − (E∩S) − (S∩M) + (M∩E∩S)

Let the number of students who enrolled in exactly 2 courses = x
So those who enrolled in all 3 = 7
Let the number of students who enrolled in exactly one course = y

Then, the total number of course enrollments =
1 × y + 2 × x + 3 × 7 = M + E + S = 47 + 47 + 57 = 151

⇒ y + 2x + 21 = 151
⇒ y + 2x = 130 — (1)

Also, total students = 100 = y + x + 7
⇒ y + x = 93 — (2)

Subtracting (2) from (1):

(y + 2x) − (y + x) = 130 − 93
⇒ x = 37

Now, from (2): y + 37 = 93 ⇒ y = 56

✅ Final Answer:

56 students enrolled in exactly one course.


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