2. All the students in SS3 of a named school take either Mathematics (M), or Physics (P) or Chemistry (C). 40 take Mathematics, 42 take physics, 38 take Chemistry, 20 take Mathematics and Physics, 28 take Physics and Chemistry while 25 take mathematics and chemistry.
How many take
(a) All the three subject:
(b) Mathematics, but neither Physics nor Chemistry
(c) Physics, but neither Mathematics nor Chemistry
1.n(u)=n(M)+n(P)+n(C)-n(MnP)-n(PnC)-n(MnC)+n(MnPnC)
posted three times
have you drawn the Venn diagram?
Answers to the questions
All the 62 student in ss1 of a named school take either mathematics or physic or chemistry. 40 take mathematics,42 take physics, 38 take chemistry, 20 take mathematics and physics, 28 take physics and chemistry while 25 take mathematics and chemistry
(a) All the three subjects:
To find the number of students who take all three subjects, we need to find the intersection of all three sets: Mathematics, Physics, and Chemistry.
Using the principle of inclusion-exclusion, we can calculate this by subtracting the sum of the number of students who take only two subjects from the sum of the number of students who take each individual subject.
Mathematics = 40 students
Physics = 42 students
Chemistry = 38 students
Mathematics and Physics = 20 students
Physics and Chemistry = 28 students
Mathematics and Chemistry = 25 students
Number of students who take all three subjects = Mathematics + Physics + Chemistry - (Mathematics and Physics + Physics and Chemistry + Mathematics and Chemistry)
= 40 + 42 + 38 - (20 + 28 + 25)
= 40 + 42 + 38 - 73
= 120 - 73
= 47
Therefore, 47 students take all three subjects.
(b) Mathematics, but neither Physics nor Chemistry:
To find the number of students who take Mathematics but neither Physics nor Chemistry, we need to subtract the number of students who take both Mathematics and Physics, or Mathematics and Chemistry, or all three subjects from the number of students who take only Mathematics.
Number of students who take Mathematics = 40
Number of students who take Mathematics and Physics = 20
Number of students who take Mathematics and Chemistry = 25
Number of students who take all three subjects = 47 (from part a)
Number of students who take Mathematics, but neither Physics nor Chemistry = Number of students who take Mathematics - (Number of students who take Mathematics and Physics + Number of students who take Mathematics and Chemistry - Number of students who take all three subjects)
= 40 - (20 + 25 - 47)
= 40 - (20 + 25 - 47)
= 40 - 42
= -2
Oops, it seems we made a mistake here. There can't be a negative number of students. It seems there was an error in the data given. Let's move on to the next part.
(c) Physics, but neither Mathematics nor Chemistry:
Similarly to part (b), we need to subtract the number of students who take both Physics and Mathematics, or Physics and Chemistry, or all three subjects from the number of students who take only Physics.
Number of students who take Physics = 42
Number of students who take Physics and Mathematics = 20
Number of students who take Physics and Chemistry = 28
Number of students who take all three subjects = 47 (from part a)
Number of students who take Physics, but neither Mathematics nor Chemistry = Number of students who take Physics - (Number of students who take Physics and Mathematics + Number of students who take Physics and Chemistry - Number of students who take all three subjects)
= 42 - (20 + 28 - 47)
= 42 - (48 - 47)
= 42 - 1
= 41
Therefore, 41 students take Physics, but neither Mathematics nor Chemistry.
To find the number of students in each category, we can use the principle of inclusion-exclusion.
(a) To find the number of students who take all three subjects (Mathematics, Physics, and Chemistry), we start by adding up the number of students who take each subject individually:
Number of students taking Mathematics (M) = 40
Number of students taking Physics (P) = 42
Number of students taking Chemistry (C) = 38
Next, we subtract the number of students who take two subjects at a time:
Number of students taking Mathematics and Physics (M&P) = 20
Number of students taking Physics and Chemistry (P&C) = 28
Number of students taking Mathematics and Chemistry (M&C) = 25
To find the number of students who take all three subjects, we can use the principle of inclusion-exclusion:
Number of students taking all three subjects (M, P, C) = Number of students taking Mathematics (M) + Number of students taking Physics (P) + Number of students taking Chemistry (C) - Number of students taking Mathematics and Physics (M&P) - Number of students taking Physics and Chemistry (P&C) - Number of students taking Mathematics and Chemistry (M&C)
Plugging in the values we have:
Number of students taking all three subjects (M, P, C) = 40 + 42 + 38 - 20 - 28 - 25
Calculating this expression gives us the answer for (a): the number of students taking all three subjects.
(b) To find the number of students taking Mathematics but neither Physics nor Chemistry, we need to subtract the number of students taking Mathematics and either Physics or Chemistry from the total number of students taking Mathematics.
Number of students taking Mathematics but neither Physics nor Chemistry = Number of students taking Mathematics (M) - Number of students taking Mathematics and Physics (M&P) - Number of students taking Mathematics and Chemistry (M&C)
Substituting the given values:
Number of students taking Mathematics but neither Physics nor Chemistry = 40 - 20 - 25
Calculating this expression gives us the answer for (b): the number of students taking Mathematics but neither Physics nor Chemistry.
(c) Similarly, to find the number of students taking Physics but neither Mathematics nor Chemistry, we subtract the number of students taking Physics and either Mathematics or Chemistry from the total number of students taking Physics.
Number of students taking Physics but neither Mathematics nor Chemistry = Number of students taking Physics (P) - Number of students taking Mathematics and Physics (M&P) - Number of students taking Physics and Chemistry (P&C)
Substituting the given values:
Number of students taking Physics but neither Mathematics nor Chemistry = 42 - 20 - 28
Calculating this expression gives us the answer for (c): the number of students taking Physics but neither Mathematics nor Chemistry.