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In a certain class of 66 students, 30 students study Science, 40 students study Humanities and 40 students study Dance. If all the students studied at least one of the subjects, then what is the maximum possible number of students who study all the three subjects?
- a)38
- b)35
- c)32
- d)22
- e)20
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
In a certain class of 66 students, 30 students study Science, 40 stude...
Understanding the Problem
In a class of 66 students:
- 30 study Science
- 40 study Humanities
- 40 study Dance
All students study at least one subject.
Using the Principle of Inclusion-Exclusion
To find the maximum number of students studying all three subjects, we can use the principle of inclusion-exclusion.
1. Total Students: 66
2. Students Studying at least One Subject: 66 (given)
3. Students Studying Only One or Two Subjects:
- Total enrolled in Science, Humanities, and Dance:
- Science + Humanities + Dance = 30 + 40 + 40 = 110
4. Double Counting: Since students can study more than one subject, the sum of students studying each subject exceeds the total. Let's denote the number of students studying all three subjects as x.
Calculating Overlap
- Total number of students studying at least one subject must equal the total actual students:
66 = (30 - y) + (40 - z) + (40 - w) + x
Here, y, z, and w represent students studying exactly two subjects.
- The maximum overlap occurs when the number of students studying two subjects and all three subjects is maximized.
Finding x
To calculate the maximum x:
- If we assume the minimum number of students studying only two subjects (which is zero), we can rearrange the equation:
66 = 110 - (y + z + w) + x
- Simplifying gives us:
y + z + w = 110 - 66 + x
y + z + w = 44 + x
To maximize x, we need y + z + w to be minimized. If we set y + z + w = 0, then:
44 + x = 0 => x = 22
Conclusion
Thus, the maximum possible number of students who study all three subjects is 22. Therefore, the correct answer is option 'D'.
In a class of 66 students:
- 30 study Science
- 40 study Humanities
- 40 study Dance
All students study at least one subject.
Using the Principle of Inclusion-Exclusion
To find the maximum number of students studying all three subjects, we can use the principle of inclusion-exclusion.
1. Total Students: 66
2. Students Studying at least One Subject: 66 (given)
3. Students Studying Only One or Two Subjects:
- Total enrolled in Science, Humanities, and Dance:
- Science + Humanities + Dance = 30 + 40 + 40 = 110
4. Double Counting: Since students can study more than one subject, the sum of students studying each subject exceeds the total. Let's denote the number of students studying all three subjects as x.
Calculating Overlap
- Total number of students studying at least one subject must equal the total actual students:
66 = (30 - y) + (40 - z) + (40 - w) + x
Here, y, z, and w represent students studying exactly two subjects.
- The maximum overlap occurs when the number of students studying two subjects and all three subjects is maximized.
Finding x
To calculate the maximum x:
- If we assume the minimum number of students studying only two subjects (which is zero), we can rearrange the equation:
66 = 110 - (y + z + w) + x
- Simplifying gives us:
y + z + w = 110 - 66 + x
y + z + w = 44 + x
To maximize x, we need y + z + w to be minimized. If we set y + z + w = 0, then:
44 + x = 0 => x = 22
Conclusion
Thus, the maximum possible number of students who study all three subjects is 22. Therefore, the correct answer is option 'D'.
Community Answer
In a certain class of 66 students, 30 students study Science, 40 stude...
Given:
- Total number of students = 66
- Students who take:
- Science = 30
- Humanities = 40
- Dance = 40
- None of these 3 subjects = 0
- Maximum possible number of students who study all 3 subjects = x
To find: X – N = ?
Approach:
- To answer the question, we need to find the value of x
- We’ll visually represent the given information to analyze the maximum possible overlap for 3 subjects.
Working Out:
The shaded part represents the students who study all three subjects: Science, Humanities, and Dance.
In the problem, the shaded part is labeled as x, which represents the maximum number of students who study all three subjects.
- Analyzing the Maximum possible overlap for the 3 subject
- To maximize x, i.e. to minimize the other overlap regions, among the sets, let’s consider the number of students who study only two subjects to be 0
- So, number of students who study only Science = 30 – x
- Number of students who study only Humanities = 40 -x
- Number of students who study only Dance = 40 – x
- So, number of students who study only 1 subject + Number of students who study all the 3 subjects = 66
- (30-x) + (40-x) + (40- x) + x = 66
- 2x = 44
- x = 22
- Getting to the answer
- So, the maximum number of students who can study all the 3 subjects is 22
Looking at the answer choices, we see that the correct answer is Option D
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In a certain class of 66 students, 30 students study Science, 40 students study Humanities and 40 students study Dance. If all the students studied at least one of the subjects, then what is the maximum possible number of students who study all the three subjects?a)38b)35c)32d)22e)20Correct answer is option 'D'. Can you explain this answer?
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In a certain class of 66 students, 30 students study Science, 40 students study Humanities and 40 students study Dance. If all the students studied at least one of the subjects, then what is the maximum possible number of students who study all the three subjects?a)38b)35c)32d)22e)20Correct answer is option 'D'. Can you explain this answer? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In a certain class of 66 students, 30 students study Science, 40 students study Humanities and 40 students study Dance. If all the students studied at least one of the subjects, then what is the maximum possible number of students who study all the three subjects?a)38b)35c)32d)22e)20Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a certain class of 66 students, 30 students study Science, 40 students study Humanities and 40 students study Dance. If all the students studied at least one of the subjects, then what is the maximum possible number of students who study all the three subjects?a)38b)35c)32d)22e)20Correct answer is option 'D'. Can you explain this answer?.
In a certain class of 66 students, 30 students study Science, 40 students study Humanities and 40 students study Dance. If all the students studied at least one of the subjects, then what is the maximum possible number of students who study all the three subjects?a)38b)35c)32d)22e)20Correct answer is option 'D'. Can you explain this answer? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In a certain class of 66 students, 30 students study Science, 40 students study Humanities and 40 students study Dance. If all the students studied at least one of the subjects, then what is the maximum possible number of students who study all the three subjects?a)38b)35c)32d)22e)20Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a certain class of 66 students, 30 students study Science, 40 students study Humanities and 40 students study Dance. If all the students studied at least one of the subjects, then what is the maximum possible number of students who study all the three subjects?a)38b)35c)32d)22e)20Correct answer is option 'D'. Can you explain this answer?.
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