53
Applying Inclusion-Exclusion Principle
We need to find the number of students taking all three subjects: Chemistry (C), Physics (P), and Mathematics (M).
We are given:
- Total students, $|C \cup P \cup M| = 500$
- Number taking Chemistry, $|C| = 329$
- Number taking Physics, $|P| = 186$
- Number taking Mathematics, $|M| = 295$
- Number taking Chemistry and Physics, $|C \cap P| = 83$
- Number taking Chemistry and Mathematics, $|C \cap M| = 217$
- Number taking Physics and Mathematics, $|P \cap M| = 63$
The Principle of Inclusion-Exclusion for three sets states:
$|C \cup P \cup M| = |C| + |P| + |M| - |C \cap P| - |C \cap M| - |P \cap M| + |C \cap P \cap M|$Calculating Students in All 3 Subjects
Let $x$ represent the number of students taking all three subjects, i.e., $x = |C \cap P \cap M|$. Plugging the given values into the formula:
$500 = 329 + 186 + 295 - 83 - 217 - 63 + x$First, sum the enrollments in individual subjects:
$329 + 186 + 295 = 810$Next, sum the enrollments in pairs of subjects:
$83 + 217 + 63 = 363$Substitute these sums back into the equation:
$500 = 810 - 363 + x$ $500 = 447 + x$Solve for $x$:
$x = 500 - 447$ $x = 53$Therefore, 53 students are taking all three subjects.
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?
