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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?
- a)37
- b)43
- c)47
- d)53
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
500 students are taking one or more courses out of Chemistry, Physics,...
To solve this problem, we can use the principle of inclusion-exclusion. We have the number of students enrolled in each subject, as well as the number of students enrolled in each combination of subjects.
Let's break down the problem step by step:
1. Find the number of students taking at least one subject:
- Number of students taking Chemistry = 329
- Number of students taking Physics = 186
- Number of students taking Mathematics = 295
To find the total number of students taking at least one subject, we can add the individual counts and subtract the counts of students taking two or more subjects (to avoid double counting):
Total = Chemistry + Physics + Mathematics - (Chemistry and Physics) - (Chemistry and Mathematics) - (Physics and Mathematics)
Total = 329 + 186 + 295 - 83 - 217 - 63
Total = 447
Therefore, there are 447 students taking at least one subject.
2. Find the number of students taking exactly two subjects:
- Number of students taking Chemistry and Physics = 83
- Number of students taking Chemistry and Mathematics = 217
- Number of students taking Physics and Mathematics = 63
To find the total number of students taking exactly two subjects, we can add the counts of students taking two subjects:
Total = (Chemistry and Physics) + (Chemistry and Mathematics) + (Physics and Mathematics)
Total = 83 + 217 + 63
Total = 363
Therefore, there are 363 students taking exactly two subjects.
3. Find the number of students taking all three subjects:
To find the number of students taking all three subjects, we can subtract the number of students taking at least one subject from the total number of students:
Number of students taking all three subjects = Total - (Chemistry + Physics + Mathematics)
Number of students taking all three subjects = 447 - (329 + 186 + 295)
Number of students taking all three subjects = 447 - 810
Number of students taking all three subjects = -363
However, it is not possible to have a negative number of students. This means that there is an error in the information given or the calculations made. Please double-check the given data and calculations to determine the correct answer.
Let's break down the problem step by step:
1. Find the number of students taking at least one subject:
- Number of students taking Chemistry = 329
- Number of students taking Physics = 186
- Number of students taking Mathematics = 295
To find the total number of students taking at least one subject, we can add the individual counts and subtract the counts of students taking two or more subjects (to avoid double counting):
Total = Chemistry + Physics + Mathematics - (Chemistry and Physics) - (Chemistry and Mathematics) - (Physics and Mathematics)
Total = 329 + 186 + 295 - 83 - 217 - 63
Total = 447
Therefore, there are 447 students taking at least one subject.
2. Find the number of students taking exactly two subjects:
- Number of students taking Chemistry and Physics = 83
- Number of students taking Chemistry and Mathematics = 217
- Number of students taking Physics and Mathematics = 63
To find the total number of students taking exactly two subjects, we can add the counts of students taking two subjects:
Total = (Chemistry and Physics) + (Chemistry and Mathematics) + (Physics and Mathematics)
Total = 83 + 217 + 63
Total = 363
Therefore, there are 363 students taking exactly two subjects.
3. Find the number of students taking all three subjects:
To find the number of students taking all three subjects, we can subtract the number of students taking at least one subject from the total number of students:
Number of students taking all three subjects = Total - (Chemistry + Physics + Mathematics)
Number of students taking all three subjects = 447 - (329 + 186 + 295)
Number of students taking all three subjects = 447 - 810
Number of students taking all three subjects = -363
However, it is not possible to have a negative number of students. This means that there is an error in the information given or the calculations made. Please double-check the given data and calculations to determine the correct answer.
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Community Answer
500 students are taking one or more courses out of Chemistry, Physics,...
Method 1
There are 500 students in total.
Only Chemistry: 329 - ( 217 + 83 ) = 29
Only Mathematics : 295 - ( 217 + 63 ) = 15
Only Physics : 186 - ( 83 + 63 ) = 40
Finally :
500 - ( 29 + 15 + 40 + 83 + 63 + 217 ) = 500 - 447 = 53 students.
Method 2
Total number of students
= n(P) + n(C) + n(M) − n(P∩C) − n(P∩M)− n(C∩M) + n(P∩C∩M)
⇒ 500 =329 + 295 + 186 − 217 − 83 − 63 + x
⇒ x = 53.
There are 500 students in total.
Only Chemistry: 329 - ( 217 + 83 ) = 29
Only Mathematics : 295 - ( 217 + 63 ) = 15
Only Physics : 186 - ( 83 + 63 ) = 40
Finally :
500 - ( 29 + 15 + 40 + 83 + 63 + 217 ) = 500 - 447 = 53 students.
Method 2
Total number of students
= n(P) + n(C) + n(M) − n(P∩C) − n(P∩M)− n(C∩M) + n(P∩C∩M)
⇒ 500 =329 + 295 + 186 − 217 − 83 − 63 + x
⇒ x = 53.
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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?a)37b)43c)47d)53Correct answer is option 'D'. Can you explain this answer?
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