This problem involves finding the total number of students in a group based on their preferences for Mathematics and English.
Calculating Total Students Using Set Theory
We can solve this using the principle of inclusion-exclusion for sets.
Given Information:
- Number of students who like Mathematics ($ \text{Math} $): 10
- Number of students who like English ($ \text{Eng} $): 12
- Number of students who like both Mathematics and English ($ \text{Math} \cap \text{Eng} $): 4
- Number of students who like neither Mathematics nor English ($ \text{Neither} $): 6
Step 1: Calculate the number of students who like at least one subject.
Use the formula for the union of two sets:
$ |\text{Math} \cup \text{Eng}| = |\text{Math}| + |\text{Eng}| - |\text{Math} \cap \text{Eng}| $Substitute the given values:
$ |\text{Math} \cup \text{Eng}| = 10 + 12 - 4 $ $ |\text{Math} \cup \text{Eng}| = 22 - 4 $ $ |\text{Math} \cup \text{Eng}| = 18 $So, 18 students like at least Mathematics or English or both.
Step 2: Calculate the total number of students in the group.
The total number of students is the sum of those who like at least one subject and those who like neither subject.
$ \text{Total Students} = |\text{Math} \cup \text{Eng}| + \text{Neither} $Substitute the values calculated:
$ \text{Total Students} = 18 + 6 $ $ \text{Total Students} = 24 $Conclusion:
Therefore, the total number of students in the group is 24.
Important Questions from Venn Diagrams
Each row of Column-I has three items and each item is represented by a circle in Column-II. The arrangement of circles in Column-II represents the relationship among the items in Column-I.
Identify the option that has the most appropriate match between Column-I and Column-II.
Note: The figure shown are representative.
Column-I Column-II (1) Animal, Zebra, Giraffe (P) 
(2) Director, Producer, Actor (Q) 
(3) Word, Sentence, Novel (R) 
(4) Pianist, Guitarist, Instrumentalist (S) 
- To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly. A total of 6,30,000 candidates appeared for the test. Question A was correctly answered by 3,30,000 candidates. Question B was answered correctly by 2,50,000 candidates. Question C was answered correctly by 2,60,000 candidates. Both questions A and B were answered correctly by 1,00,000 candidates. Both questions B and C were answered correctly by 90,000 candidates. Both questions A and C were answered correctly by 80,000 candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test?
- Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films?
- 500 students are taking one or more courses out of Chemistry, Physics, and Mathematics. Registration records indicate course enrolment as follows: Chemistry (329), Physics (186), Mathematics (295), Chemistry and Physics (83), Chemistry and Mathematics (217), and Physics and Mathematics (63). How many students are taking all 3 subjects?
In the given diagram, teachers are represented in the triangle, researchers in the circle and administrators in the rectangle. Out of the total number of the people, the percentage of administrators shall be in the range of ___________.
