Venn Diagrams

The Venn diagram shows the relationship between three categories: Doctors (represented by the triangle), Professors (represented by the circle), and Married People (represented by the rectangle). The numbers in the different regions indicate the individuals belonging to specific combinations of these categories. We need to find the number that indicates doctors who are not married.
The triangle represents Doctors. The numbers within the triangle are 1, 5, 6, and (B) These represent individuals who are doctors.
The rectangle represents Married People. The numbers within the rectangle are 5, 6, 2, and (C) These represent individuals who are married. Doctors who are married are represented by the region where the triangle and the rectangle overlap. This region contains the numbers 5, 6, and (B) Doctors who are not married are represented by the region within the triangle but outside the rectangle. This region contains the number (A) Therefore, the number that indicates doctors who are not married is (A)

The Venn diagram shows the number of students who failed in different combinations of English, Hindi, and Maths. We need to find the percentage of students who failed in at least two subjects. This means we need to count the number of students who failed in exactly two subjects or in all three subjects. From the Venn diagram:
Students who failed in English and Hindi only: 10
Students who failed in Hindi and Maths only: 12
Students who failed in English and Maths only: 12
Students who failed in English, Hindi, and Maths: 5 The number of students who failed in at least two subjects is the sum of those who failed in exactly two subjects and those who failed in all three subjects: Number of students failed in at least two subjects Number of students failed in at least two subjects The total number of candidates who appeared in the examination is 500. The percentage of students who failed in at least two subjects is: $ $ Therefore, the percentage of students who failed in at least two subjects is .

In real life, some teachers may also be writers, some teachers may be dancers, and some writers may be dancers. However, none of these categories is completely contained within another. The three sets overlap partially, making option (a) with intersecting circles the best representation. This Venn diagram captures all possible overlaps among the three professions.