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
- 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?
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 ___________.
Out of \(100\) textile companies, \(10\) companies are involved in spinning, weaving and chemical processing, \(25\) companies are involved in spinning and chemical processing, and \(30\) companies are involved in weaving and chemical processing. If \(65\) companies are involved in chemical processing, the number of companies involved {ONLY} in chemical processing is ________________________.
- In a class of 300 students in an M.Tech programme, each student is required to take at least one subject from the following three:
M600: Advanced Engineering Mathematics
C600: Computational Methods for Engineers
E600: Experimental Techniques for Engineers
The registration data for the M.Tech class shows that 100 students have taken M600, 200 students have taken C600, and 60 students have taken E600. What is the maximum possible number of students in the class who have taken all the above three subjects? The Venn Diagram below shows numbers of species in three forest types. Which of the following statements is true?
