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From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations?
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From 50 students taking examinations in mathematics, physics, chemistr...
Given information:
- There are 50 students taking examinations in mathematics, physics, and chemistry.
- Each student has passed in at least one of the subjects.
- 37 students passed mathematics.
- 24 students passed physics.
- 43 students passed chemistry.
- At most 19 students passed mathematics and physics.
- At most 29 students passed mathematics and chemistry.
- At most 20 students passed physics and chemistry.
Objective:
- Find the largest possible number of students who could have passed all three examinations.
Solution:
Step 1: Analyzing the given information
Let's analyze the given information to understand the relationships between the number of students who have passed each subject.
- We are given that 37 students passed mathematics, so at least 37 students have passed at least one subject.
- Similarly, 24 students passed physics, and 43 students passed chemistry.
- At most 19 students passed mathematics and physics, at most 29 students passed mathematics and chemistry, and at most 20 students passed physics and chemistry.
Step 2: Determining the minimum number of students who have passed all three subjects
To determine the minimum number of students who have passed all three subjects, we need to find the intersection of the three given sets:
- Let A represent the set of students who passed mathematics.
- Let B represent the set of students who passed physics.
- Let C represent the set of students who passed chemistry.
From the given information:
- |A| = 37 (number of students who passed mathematics)
- |B| = 24 (number of students who passed physics)
- |C| = 43 (number of students who passed chemistry)
- |A ∩ B| ≤ 19 (at most 19 students passed mathematics and physics)
- |A ∩ C| ≤ 29 (at most 29 students passed mathematics and chemistry)
- |B ∩ C| ≤ 20 (at most 20 students passed physics and chemistry)
To determine the minimum number of students who have passed all three subjects, we can find the maximum value of |A ∩ B ∩ C| using the principle of inclusion-exclusion.
Step 3: Applying the principle of inclusion-exclusion
The principle of inclusion-exclusion states that:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
We are interested in finding |A ∩ B ∩ C|, so we rearrange the equation:
|A ∩ B ∩ C| = |A| + |B| + |C| - |A ∪ B ∪ C| + |A ∩ B| + |A ∩ C| + |B ∩ C|
We know the values of |A|, |B|, |C|, |A ∩ B|, |A ∩ C|, and |B ∩ C|, so we can substitute them into the equation:
|A ∩ B ∩ C| = 37 + 24 + 43 - 50 + |A ∩ B| + |A ∩ C| + |B ∩ C
- There are 50 students taking examinations in mathematics, physics, and chemistry.
- Each student has passed in at least one of the subjects.
- 37 students passed mathematics.
- 24 students passed physics.
- 43 students passed chemistry.
- At most 19 students passed mathematics and physics.
- At most 29 students passed mathematics and chemistry.
- At most 20 students passed physics and chemistry.
Objective:
- Find the largest possible number of students who could have passed all three examinations.
Solution:
Step 1: Analyzing the given information
Let's analyze the given information to understand the relationships between the number of students who have passed each subject.
- We are given that 37 students passed mathematics, so at least 37 students have passed at least one subject.
- Similarly, 24 students passed physics, and 43 students passed chemistry.
- At most 19 students passed mathematics and physics, at most 29 students passed mathematics and chemistry, and at most 20 students passed physics and chemistry.
Step 2: Determining the minimum number of students who have passed all three subjects
To determine the minimum number of students who have passed all three subjects, we need to find the intersection of the three given sets:
- Let A represent the set of students who passed mathematics.
- Let B represent the set of students who passed physics.
- Let C represent the set of students who passed chemistry.
From the given information:
- |A| = 37 (number of students who passed mathematics)
- |B| = 24 (number of students who passed physics)
- |C| = 43 (number of students who passed chemistry)
- |A ∩ B| ≤ 19 (at most 19 students passed mathematics and physics)
- |A ∩ C| ≤ 29 (at most 29 students passed mathematics and chemistry)
- |B ∩ C| ≤ 20 (at most 20 students passed physics and chemistry)
To determine the minimum number of students who have passed all three subjects, we can find the maximum value of |A ∩ B ∩ C| using the principle of inclusion-exclusion.
Step 3: Applying the principle of inclusion-exclusion
The principle of inclusion-exclusion states that:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
We are interested in finding |A ∩ B ∩ C|, so we rearrange the equation:
|A ∩ B ∩ C| = |A| + |B| + |C| - |A ∪ B ∪ C| + |A ∩ B| + |A ∩ C| + |B ∩ C|
We know the values of |A|, |B|, |C|, |A ∩ B|, |A ∩ C|, and |B ∩ C|, so we can substitute them into the equation:
|A ∩ B ∩ C| = 37 + 24 + 43 - 50 + |A ∩ B| + |A ∩ C| + |B ∩ C
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From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations?
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From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations?.
From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations?.
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From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations?, a detailed solution for From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations? has been provided alongside types of From 50 students taking examinations in mathematics, physics, chemistry,each of the students has passed in at least one of the subject.37passed mathematics,24physics and 43chemistry .At most 19 passed mathematics and physics ,at most 29 mathematics and chemistry and at most 20 physics and chemistry.Then the largest possible number that could have passed all these examinations? theory, EduRev gives you an
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